| تعداد نشریات | 8 |
| تعداد شمارهها | 439 |
| تعداد مقالات | 5,658 |
| تعداد مشاهده مقاله | 7,721,375 |
| تعداد دریافت فایل اصل مقاله | 6,358,103 |
A Unified Gram-Schmidt–Ritz Solution for Vibration Analysis of Nanoplates with Elastic Boundary Conditions | ||
| AUT Journal of Mechanical Engineering | ||
| دوره 9، شماره 4، دی 2025، صفحه 413-430 اصل مقاله (1.56 M) | ||
| نوع مقاله: Research Article | ||
| شناسه دیجیتال (DOI): 10.22060/ajme.2025.24153.6178 | ||
| نویسنده | ||
| Hossein Pakdaman* | ||
| Department of Civil Engineering, K.N. Toosi University of Technology, Valiasr Ave., Tehran, Iran | ||
| چکیده | ||
| A novel and unified approach is presented for analyzing the free vibration of rectangular nanoplates with elastic boundary conditions. The theoretical modeling is achieved using the nonlocal Mindlin plate theory, which accounts for the size-dependent behavior of nanoplates, while the artificial spring technique is employed to accommodate a wide range of boundary conditions, including classical boundary conditions, elastic boundary conditions, and their combinations. The governing equations of motion are derived using the virtual displacement principle, followed by the application of the weighted residual method to obtain the nonlocal quadratic functional. The Rayleigh-Ritz method, employing Gram-Schmidt polynomial series as the admissible displacement functions, is then utilized to solve the eigenvalue problems associated with the free vibration of nanoplates. The present approach is validated through a series of comparison and convergence studies, which demonstrate its high accuracy and low computational cost. Finally, parametric numerical investigations are conducted to elucidate the effects of variations in spring stiffness on the natural frequencies of nanoplates. It is shown that the proposed method can easily compute the natural frequencies of nanoplates with elastic boundary conditions. | ||
| کلیدواژهها | ||
| Free vibration؛ Nanoplate؛ Gram-Schmidt Polynomial؛ Rayleigh–Ritz Method؛ Elastic Boundary Conditions | ||
| مراجع | ||
|
[1] K.E. Whitener Jr, P.E. Sheehan, Graphene synthesis, Diamond and related materials, 46 (2014) 25-34.
[2] B.K. Choi, J. Kim, Z. Luo, J. Kim, J.H. Kim, T. Hyeon, S. Mehraeen, S. Park, J. Park, Shape transformation mechanism of gold nanoplates, ACS nano, 17(3) (2023) 2007-2018.
[3] J. Chen, S. Li, Y. Chen, J. Yang, J. Dong, X. Lu, l-cysteine-terminated triangular silver nanoplates/MXene nanosheets are used as electrochemical biosensors for efficiently detecting 5-hydroxytryptamine, Analytical Chemistry, 93(49) (2021) 16655-16663.
[4] Y. Zhao, M. Zhao, X. Ding, Z. Liu, H. Tian, H. Shen, X. Zu, S. Li, L. Qiao, One-step colloid fabrication of nickel phosphides nanoplate/nickel foam hybrid electrode for high-performance asymmetric supercapacitors, Chemical Engineering Journal, 373 (2019) 1132-1143.
[5] R. Ding, S. Chen, J. Lv, W. Zhang, X.-d. Zhao, J. Liu, X. Wang, T.-j. Gui, B.-j. Li, Y.-z. Tang, Study on graphene modified organic anti-corrosion coatings: A comprehensive review, Journal of Alloys and Compounds, 806 (2019) 611-635.
[6] W. Chen, S. Luo, M. Sun, X. Wu, Y. Zhou, Y. Liao, M. Tang, X. Fan, B. Huang, Z. Quan, High‐entropy intermetallic PtRhBiSnSb nanoplates for highly efficient alcohol oxidation electrocatalysis, Advanced Materials, 34(43) (2022) 2206276.
[7] H. Hu, H. Li, Y. Lei, J. Liu, X. Liu, R. Wang, J. Peng, X. Wang, Mg-doped LiMn0. 8Fe0. 2PO4/C nano-plate as a high-performance cathode material for lithium-ion batteries, Journal of Energy Storage, 73 (2023) 109006.
[8] M. Guerroudj, A. Drai, A.A. Daikh, M.S.A. Houari, B. Aour, M.A. Eltaher, M.-O. Belarbi, Size-dependent free vibration analysis of multidirectional functionally graded nanobeams via a nonlocal strain gradient theory, Journal of Engineering Mathematics, 146 (2024) 20.
[9] I. Jafarsadeghi-Pournaki, G. Rezazadeh, R. Shabani, Nonlinear instability modeling of a nonlocal strain gradient functionally graded capacitive nanobridge in thermal environment, International Journal of Applied Mechanics, 10 (2018) 1850083.
[10] Z. Rahimi, G. Rezazadeh, W. Sumelka, A nonlocal fractional stress–strain gradient theory, International Journal of Mechanics and Materials in Design, 16 (2020) 265–278.
[11] B. Mawphlang, P. Patra, Study of the large bending behavior of CNTs using LDTM and nonlocal elasticity theory, International Journal of Non-Linear Mechanics, 166 (2024) 104828.
[12] S. Valilou, G. Rezazadeh, R. Shabani, M. Fathalilou, Bifurcation analysis of a capacitive microresonator considering nonlocal elasticity theory, International Journal of Nonlinear Sciences and Numerical Simulation, 15 (2014) 241–249.
[13] G. Rezazadeh, M. Sheikhlou, R. Shabani, Analysis of bias DC voltage effect on thermoelastic damping ratio in short nanobeam resonators based on nonlocal elasticity theory and dual-phase-lagging heat conduction model, Meccanica, 50 (2015) 2963–2976.
[14] M. Al-Furjan, M. Xu, A. Farrokhian, G.S. Jafari, X. Shen, R. Kolahchi, On wave propagation in piezoelectric-auxetic honeycomb-2D-FGM micro-sandwich beams based on modified couple stress and refined zigzag theories, Waves in Random and Complex Media, 35(1) (2025) 1147-1171.
[15] S. Dastjerdi, F. Naeijian, B. Akgöz, Ö. Civalek, On the mechanical analysis of microcrystalline cellulose sheets, International Journal of Engineering Science, 166 (2021) 103500.
[16] Ş.D. Akbaş, S. Dastjerdi, B. Akgöz, et al., Dynamic analysis of functionally graded porous microbeams under moving load, Transport in Porous Media, 142 (2022) 209–227.
[17] M. Davoodi Yekta, A. Rahi, Design of two layer clamped-clamped microsensor based on classical and non-classical theories, AUT Journal of Mechanical Engineering, 9(1) (2025) 19-32.
[18] A.C. Eringen, Microcontinuum field theories: I. Foundations and solids, Springer Science & Business Media, 2012.
[19] J. Peddieson, G.R. Buchanan, R.P. McNitt, Application of nonlocal continuum models to nanotechnology, International journal of engineering science, 41(3-5) (2003) 305-312.
[20] K. Kiani, H. Pakdaman, Bilaterally nonlocal dynamics of layer-by-layer assembly of double-walled carbon nanotubes accounting for intertube rigorous van der Waals forces, European Journal of Mechanics-A/Solids, 80 (2020) 103876.
[21] K. Kiani, H. Pakdaman, Nonlocal vibrations and potential instability of monolayers from double-walled carbon nanotubes subjected to temperature gradients, International Journal of Mechanical Sciences, 144 (2018) 576-599.
[22] H. Azimloo, G. Rezazadeh, R. Shabani, Bifurcation analysis of an electrostatically actuated nanobeam based on nonlocal theory considering centrifugal forces, International Journal of Nonlinear Sciences and Numerical Simulation, 21 (2020) 303–318.
[23] D.M. Tien, D.V. Thom, P.V. Minh, N.C. Tho, T.N. Doan, D.N. Mai, The application of the nonlocal theory and various shear strain theories for bending and free vibration analysis of organic nanoplates, Mechanics Based Design of Structures and Machines, 52(1) (2024) 588-610.
[24] K. Kiani, H. Pakdaman, Three-dimensional vibrations and instabilities of electron-transporting multi-layered graphene sheets via nonlocal-continuum-based models, Applied Mathematical Modelling, 145 (2025) 116103.
[25] Ö. Civalek, B. Uzun, M.Ö. Yayli, Torsional and longitudinal vibration analysis of a porous nanorod with arbitrary boundaries, Physica B: Condensed Matter, 633 (2022) 413761.
[26] H. Pakdaman, M. Roshan, S. Soltani, Vibrational analysis of two crossed graphene nanoribbons via nonlocal differential/integral models, Acta Mechanica, 235(2) (2024) 797-818.
[27] F. Khosravi, S.A. Hosseini, B.A. Hamidi, On torsional vibrations of triangular nanowire, Thin-Walled Structures, 148 (2020) 106591.
[28] S. Foroutan, A. Haghshenas, M. Hashemian, S.A. Eftekhari, D. Toghraie, Spatial buckling analysis of current-carrying nanowires in the presence of a longitudinal magnetic field accounting for both surface and nonlocal effects, Physica E: Low-dimensional Systems and Nanostructures, 97 (2018) 191-205.
[29] P. Lu, P. Zhang, H. Lee, C. Wang, J. Reddy, Non-local elastic plate theories, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463(2088) (2007) 3225-3240.
[30] J. Phadikar, S. Pradhan, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Computational materials science, 49(3) (2010) 492-499.
[31] Q.H. Pham, V.K. Tran, T.T. Tran, V.C. Nguyen, A.M. Zenkour, Nonlocal higher-order finite element modeling for vibration analysis of viscoelastic orthotropic nanoplates resting on variable viscoelastic foundation, Composite Structures, 318 (2023) 117067.
[32] R.A. Arpanahi, B. Mohammadi, M.T. Ahmadian, S.H. Hashemi, Study on the buckling behavior of nonlocal nanoplate submerged in viscous moving fluid, International Journal of Dynamics and Control, 11(6) (2023) 2820-2830.
[33] F. Ebrahimi, N. Shafiei, M. Kazemi, S.M. Mousavi Abdollahi, Thermo-mechanical vibration analysis of rotating nonlocal nanoplates applying generalized differential quadrature method, Mechanics of Advanced Materials and Structures, 24(15) (2017) 1257-1273.
[34] S. Dastjerdi, B. Akgöz, New static and dynamic analyses of macro and nano FGM plates using exact three-dimensional elasticity in thermal environment, Composite Structures, 192 (2018) 626-641.
[35] S. Chakraverty, L. Behera, Free vibration of rectangular nanoplates using Rayleigh–Ritz method, Physica E: Low-dimensional Systems and Nanostructures, 56 (2014) 357-363.
[36] H. Analooei, M. Azhari, A. Heidarpour, Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method, Applied Mathematical Modelling, 37(10-11) (2013) 6703-6717.
[37] H. Tanzadeh, H. Amoushahi, Buckling analysis of orthotropic nanoplates based on nonlocal strain gradient theory using the higher-order finite strip method (H-FSM), European Journal of Mechanics-A/Solids, 95 (2022) 104622.
[38] S.S. Ma’en, W.G. Al-Kouz, Vibration analysis of non-uniform orthotropic Kirchhoff plates resting on elastic foundation based on nonlocal elasticity theory, International Journal of Mechanical Sciences, 114 (2016) 1-11.
[39] Y. Zhang, L. Zhang, K. Liew, J. Yu, Buckling analysis of graphene sheets embedded in an elastic medium based on the kp-Ritz method and non-local elasticity theory, Engineering Analysis with Boundary Elements, 70 (2016) 31-39.
[40] Ö. Civalek, B. Akgöz, Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix, Computational Materials Science, 77 (2013) 295-303.
[41] S.H. Hashemi, H. Mehrabani, A. Ahmadi-Savadkoohi, Forced vibration of nanoplate on viscoelastic substrate with consideration of structural damping: An analytical solution, Composite Structures, 133 (2015) 8-15.
[42] S. Pradhan, J. Phadikar, Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration, 325(1-2) (2009) 206-223.
[43] M. Panyatong, B. Chinnaboon, S. Chucheepsakul, Free vibration analysis of FG nanoplates embedded in elastic medium based on second-order shear deformation plate theory and nonlocal elasticity, Composite Structures, 153 (2016) 428-441.
[44] R. Aghababaei, J. Reddy, Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates, Journal of Sound and Vibration, 326(1-2) (2009) 277-289.
[45] M. Ilkhani, A. Bahrami, S. Hosseini-Hashemi, Free vibrations of thin rectangular nano-plates using wave propagation approach, Applied Mathematical Modelling, 40(2) (2016) 1287-1299.
[46] D. Rong, J. Fan, C. Lim, X. Xu, Z. Zhou, A new analytical approach for free vibration, buckling and forced vibration of rectangular nanoplates based on nonlocal elasticity theory, International Journal of Structural Stability and Dynamics, 18(04) (2018) 1850055.
[47] S. Hosseini-Hashemi, M. Zare, R. Nazemnezhad, An exact analytical approach for free vibration of Mindlin rectangular nano-plates via nonlocal elasticity, Composite Structures, 100 (2013) 290-299.
[48] S. Hosseini-Hashemi, M. Kermajani, R. Nazemnezhad, An analytical study on the buckling and free vibration of rectangular nanoplates using nonlocal third-order shear deformation plate theory, European Journal of Mechanics-A/Solids, 51 (2015) 29-43.
[49] X. Zheng, M. Huang, D. An, C. Zhou, R. Li, New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method, Scientific Reports, 11(1) (2021) 2939.
[50] Z. Wang, Y. Xing, Q. Sun, Y. Yang, Highly accurate closed-form solutions for free vibration and eigenbuckling of rectangular nanoplates, Composite Structures, 210 (2019) 822-830.
[51] Z. Qin, S. Zhao, X. Pang, B. Safaei, F. Chu, A unified solution for vibration analysis of laminated functionally graded shallow shells reinforced by graphene with general boundary conditions, International Journal of Mechanical Sciences, 170 (2020) 105341.
[52] G. Jin, X. Ma, S. Shi, T. Ye, Z. Liu, A modified Fourier series solution for vibration analysis of truncated conical shells with general boundary conditions, Applied Acoustics, 85 (2014) 82-96.
[53] D. Zhou, Y. Cheung, F. Au, S. Lo, Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method, International Journal of Solids and Structures, 39(26) (2002) 6339-6353.
[54] H. Lin, D. Cao, A unified Gram-Schmidt-Ritz formulation for vibration and active flutter control analysis of honeycomb sandwich plate with general elastic support, Journal of Vibroengineering, 20(5) (2018) 1982-2000.
[55] Z. Qin, F. Chu, J. Zu, Free vibrations of cylindrical shells with arbitrary boundary conditions: a comparison study, International Journal of Mechanical Sciences, 133 (2017) 91-99.
[56] S. Zhao, X. Zhang, S. Zhang, B. Safaei, Z. Qin, F. Chu, A unified modeling approach for rotating flexible shaft-disk systems with general boundary and coupling conditions, International Journal of Mechanical Sciences, 218 (2022) 107073.
[57] B. Qin, R. Zhong, Q. Wu, T. Wang, Q. Wang, A unified formulation for free vibration of laminated plate through Jacobi-Ritz method, Thin-Walled Structures, 144 (2019) 106354.
[58] X. Song, G. Jin, T. Ye, S. Zhong, A formulation for turbulent-flow-induced vibration of elastic plates with general boundary conditions, International Journal of Mechanical Sciences, 205 (2021) 106602.
[59] G. Li, Y. Xing, Z. Wang, Closed-form solutions for free vibration of rectangular nonlocal Mindlin plates with arbitrary homogeneous boundary conditions, Composites Part C: Open Access, 6 (2021) 100193.
[60] C. Li, S.K. Lai, X. Yang, On the nano-structural dependence of nonlocal dynamics and its relationship to the upper limit of nonlocal scale parameter, Applied Mathematical Modelling, 69 (2019) 127-141.
[61] C.L. Dym, I.H. Shames, Solid mechanics, Springer, 1973.
[62] Y. Song, K. Xue, Q. Li, A solution method for free vibration of intact and cracked polygonal thin plates using the Ritz method and Jacobi polynomials, Journal of Sound and Vibration, 519 (2022) 116578.
[63] R.B. Bhat, Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method, Journal of sound and vibration, 102(4) (1985) 493-499.
[64] Y. Liang, Q. Han, Prediction of the nonlocal scaling parameter for graphene sheet, European Journal of Mechanics-A/Solids, 45 (2014) 153-160.
[65] S. Sahmani, A. M. Fattahi, Calibration of developed nonlocal anisotropic shear deformable plate model for uniaxial instability of 3D metallic carbon nanosheets using MD simulations, Computer Methods in Applied Mechanics and Engineering, 322 (2017) 187-207.
[66] M. Shariati, S. S. M. N. Souq, B. Azizi, Surface- and nonlocality-dependent vibrational behavior of graphene using atomistic-modal analysis, International Journal of Mechanical Sciences, 228 (2022) 107471.
[67] S. H. Madani, M. H. Sabour, M. Fadaee, Molecular dynamics simulation of vibrational behavior of annular graphene sheet: Identification of nonlocal parameter, Journal of Molecular Graphics and Modelling, 79 (2018) 264-272. | ||
|
آمار تعداد مشاهده مقاله: 629 تعداد دریافت فایل اصل مقاله: 259 |
||