Abstract
The torsional vibration is an unavoidable part of the mechanical behavior of the nanoscale materials. The nanostructures always are not circular. On the contrary, they have more often noncircular cross sections. In this paper, the size-dependent free torsional vibration of a nanowire with a triangular (equilateral triangle) cross section based on the nonlocal strain gradient theory is investigated for the first time. Three different boundary conditions such as clamped–clamped (C–C), clamped-free (C-F), and clamped–torsional spring (C-T) boundary conditions are utilized. Moreover, the first three natural frequencies of a hollow circular nanowire along with the nanowire with horizontal and vertical hollow elliptical cross sections are evaluated. The equation of motion, as well as the boundary conditions, is extracted by using Hamilton’s principle. An analytical method is employed to reduce the order of the equation of motion and ultimately solve it. The nanowire model contains the material length scale parameter and nonlocal parameter, simultaneously to take into account both stiffness-hardening and stiffness-softening effects on the vibration of the system. The effects of the nondimensional material length scale parameter and nondimensional nonlocal parameter on the first four nondimensional natural frequencies of the nanowire are assessed. The influences of ea/l and nondimensional material length scale parameter, and the effects of l/ea and nondimensional nonlocal parameter on the nondimensional responses are examined. Also, the effects of the nonlocal parameter on the first three natural frequencies of the model having circular and hollow elliptical cross sections are illustrated. The results are proved to be valid compared to another study.
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References
J.-J. Zhu, C.-X. Kan, J.-G. Wan, M. Han, G.-H. Wang, High-yield synthesis of uniform Ag nanowires with high aspect ratios by introducing the long-chain PVP in an improved polyol process. J. Nanomater. 2011, 40 (2011)
M. Liu, P. Jin, Z. Xu, D.A. Hanaor, Y. Gan, C. Chen, Two-dimensional modeling of the self-limiting oxidation in silicon and tungsten nanowires. Theor. Appl. Mech. Lett. 6(5), 195–199 (2016)
O.L. Muskens, J.G. Rivas, R.E. Algra, E.P. Bakkers, A. Lagendijk, Design of light scattering in nanowire materials for photovoltaic applications. Nano Lett. 8(9), 2638–2642 (2008)
S. Hong, H. Lee, J. Lee, J. Kwon, S. Han, Y.D. Suh, H. Cho, J. Shin, J. Yeo, S.H. Ko, Highly stretchable and transparent metal nanowire heater for wearable electronics applications. Adv. Mater. 27(32), 4744–4751 (2015)
F. Patolsky, G. Zheng, C.M. Lieber, Nanowire sensors for medicine and the life sciences. Nanomedicine 1, 51–65 (2006)
D. Reich, M. Tanase, A. Hultgren, L. Bauer, C. Chen, G. Meyer, Biological applications of multifunctional magnetic nanowires. J. Appl. Phys. 93(10), 7275–7280 (2003)
M.J. Bierman, S. Jin, Potential applications of hierarchical branching nanowires in solar energy conversion. Energy Environ. Sci. 2(10), 1050–1059 (2009)
M. Curreli, C. Li, Y. Sun, B. Lei, M.A. Gundersen, M.E. Thompson, C. Zhou, Selective functionalization of In\(_2\)O\(_3\) nanowire mat devices for biosensing applications. J. Am. Chem. Soc. 127(19), 6922–6923 (2005)
X. Feng, K. Shankar, O.K. Varghese, M. Paulose, T.J. Latempa, C.A. Grimes, Vertically aligned single crystal TiO\(_2\) nanowire arrays grown directly on transparent conducting oxide coated glass: synthesis details and applications. Nano Lett. 8(11), 3781–3786 (2008)
A.I. Persson, M.W. Larsson, S. Stenström, B.J. Ohlsson, L. Samuelson, L.R. Wallenberg, Solid-phase diffusion mechanism for GaAs nanowire growth. Nat. Mater. 3(10), 677 (2004)
S. Mathur, S. Barth, H. Shen, J.C. Pyun, U. Werner, Size-dependent photoconductance in SnO\(_2\) nanowires. Small 1(7), 713–717 (2005)
K. Peng, Y. Xu, Y. Wu, Y. Yan, S.T. Lee, J. Zhu, Aligned single-crystalline Si nanowire arrays for photovoltaic applications. Small 1(11), 1062–1067 (2005)
Y. Zhu, T. Yu, F. Cheong, X. Xu, C. Lim, V. Tan, J. Thong, C. Sow, Large-scale synthesis and field emission properties of vertically oriented CuO nanowire films. Nanotechnology 16(1), 88 (2004)
B.A. Hamidi, S.A. Hosseini, R. Hassannejad, F. Khosravi, An exact solution on gold microbeam with thermoelastic damping via generalized Green–Naghdi and modified couple stress theories. J. Therm. Stresses 43, 1–18 (2019)
A.H. Hosseini, O. Rahmani, M. Nikmehr, I.F. Golpayegani, Axial vibration of cracked nanorods embedded in elastic foundation based on a nonlocal elasticity model. Sens. Lett. 14(10), 1019–1025 (2016)
S.A. Hosseini, O. Rahmani, Bending and vibration analysis of curved FG nanobeams via nonlocal Timoshenko model. Smart Construct. Res. 2, 1–17 (2018)
O. Rahmani, S. Hosseini, I. Ghoytasi, H. Golmohammadi, Buckling and free vibration of shallow curved micro/nano-beam based on strain gradient theory under thermal loading with temperature-dependent properties. Appl. Phys. A 123(1), 4 (2017)
H. Hayati, S.A. Hosseini, O. Rahmani, Coupled twist-bending static and dynamic behavior of a curved single-walled carbon nanotube based on nonlocal theory. Microsyst. Technol. 23(7), 2393–2401 (2017)
R. Sourki, S. Hosseini, Coupling effects of nonlocal and modified couple stress theories incorporating surface energy on analytical transverse vibration of a weakened nanobeam. Eur. Phys. J. Plus 132(4), 184 (2017)
O. Rahmani, S. Norouzi, H. Golmohammadi, S. Hosseini, Dynamic response of a double, single-walled carbon nanotube under a moving nanoparticle based on modified nonlocal elasticity theory considering surface effects. Mech. Adv. Mater. Struct. 24(15), 1274–1291 (2017)
O. Rahmani, M. Shokrnia, H. Golmohammadi, S. Hosseini, Dynamic response of a single-walled carbon nanotube under a moving harmonic load by considering modified nonlocal elasticity theory. Eur. Phys. J. Plus 133(2), 42 (2018)
S. Hosseini, O. Rahmani, Exact solution for axial and transverse dynamic response of functionally graded nanobeam under moving constant load based on nonlocal elasticity theory. Meccanica 52(6), 1441–1457 (2017)
M. Namvar, E. Rezaei, S.A. Hosseini, M. Ghadiri, Experimental and analytical investigations of vibrational behavior of U-shaped atomic force microscope probe considering thermal loading and the modified couple stress theory. Eur. Phys. J. Plus 132(6), 247 (2017)
O. Rahmani, S. Hosseini, I. Ghoytasi, H. Golmohammadi, Free vibration of deep curved FG nano-beam based on modified couple stress theory. Steel Compos. Struct. 26(5), 607–620 (2018)
O. Rahmani, S.A.H. Hosseini, H. Hayati, Frequency analysis of curved nano-sandwich structure based on a nonlocal model. Mod. Phys. Lett. B 30(10), 1650136 (2016)
M. Zarepour, S. Hosseini, A. Akbarzadeh, Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen’s differential model. Appl. Math. Model. 69, 563–582 (2019)
M. Ghadiri, S. Hosseini, M. Karami, M. Namvar, In-plane and out of plane free vibration of U-shaped AFM probes based on the nonlocal elasticity. J. Solid Mech. 10(2), 285–299 (2018)
O. Rahmani, S. Asemani, S. Hosseini, Study the surface effect on the buckling of nanowires embedded in Winkler-Pasternak elastic medium based on a nonlocal theory. J. Nanostruct. 6, 87–92 (2016)
S.A. Hosseini, F. Khosravi, M. Ghadiri, Moving axial load on dynamic response of single-walled carbon nanotubes using classical, Rayleigh and Bishop rod models based on Eringen’s theory. J. Vib. Control (2019). https://doi.org/10.1177/1077546319890170
E.C. Aifantis, On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30(10), 1279–1299 (1992)
M.M. Adeli, A. Hadi, M. Hosseini, H.H. Gorgani, Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory. Eur. Phys. J. Plus 132(9), 393 (2017)
L. Li, Y. Hu, Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects. Int. J. Mech. Sci. 120, 159–170 (2017)
A. Apuzzo, R. Barretta, S. Faghidian, R. Luciano, F.M. de Sciarra, Free vibrations of elastic beams by modified nonlocal strain gradient theory. Int. J. Eng. Sci. 133, 99–108 (2018)
H. Tang, L. Li, Y. Hu, W. Meng, K. Duan, Vibration of nonlocal strain gradient beams incorporating Poisson’s ratio and thickness effects. Thin-Walled Struct. 137, 377–391 (2019)
S.S. Mirjavadi, B.M. Afshari, M.R. Barati, A. Hamouda, Transient response of porous FG nanoplates subjected to various pulse loads based on nonlocal stress–strain gradient theory. Eur. J. Mech.-A/Solids 74, 210–220 (2019)
H. Liu, Z. Lv, H. Wu, Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory. Compos. Struct. 214, 47–61 (2019)
G.-L. She, F.-G. Yuan, Y.-R. Ren, H.-B. Liu, W.-S. Xiao, Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory. Compos. Struct. 203, 614–623 (2018)
G.-F. Wang, X.-Q. Feng, Timoshenko beam model for buckling and vibration of nanowires with surface effects. J. Phys. D Appl. Phys. 42(15), 155411 (2009)
D. Dikin, X. Chen, W. Ding, G. Wagner, R. Ruoff, Resonance vibration of amorphous SiO\(_2\) nanowires driven by mechanical or electrical field excitation. J. Appl. Phys. 93(1), 226–230 (2003)
X. Dai, F. Zhu, Z. Qian, J. Yang, Electric potential and carrier distribution in a piezoelectric semiconductor nanowire in time-harmonic bending vibration. Nano Energy 43, 22–28 (2018)
S. Narendar, S. Ravinder, S. Gopalakrishnan, Strain gradient torsional vibration analysis of micro/nano rods. Int. J. Nano Dimens. 3(1), 1–17 (2012)
L. Li, Y. Hu, Torsional vibration of bi-directional functionally graded nanotubes based on nonlocal elasticity theory. Compos. Struct. 172, 242–250 (2017)
T. Murmu, S. Adhikari, C. Wang, Torsional vibration of carbon nanotube-buckyball systems based on nonlocal elasticity theory. Physica E 43(6), 1276–1280 (2011)
C.W. Lim, C. Li, J. Yu, Free torsional vibration of nanotubes based on nonlocal stress theory. J. Sound Vib. 331(12), 2798–2808 (2012)
C. Li, Torsional vibration of carbon nanotubes: comparison of two nonlocal models and a semi-continuum model. Int. J. Mech. Sci. 82, 25–31 (2014)
R. Barretta, S.A. Faghidian, F. Marotti de Sciarra, R. Penna, F.P. Pinnola, On torsion of nonlocal Lam strain gradient FG elastic beams. Compos. Struct. 233, 111550 (2020)
A. Barr, Torsional waves in uniform rods of non-circular section. J. Mech. Eng. Sci. 4(2), 127–135 (1962)
S. Christides, A. Barr, Torsional vibration of cracked beams of non-circular cross-section. Int. J. Mech. Sci. 28(7), 473–490 (1986)
R. Barretta, L. Feo, R. Luciano, Some closed-form solutions of functionally graded beams undergoing nonuniform torsion. Compos. Struct. 123, 132–136 (2015)
N. Stephen, Comparison of dynamic torsion theories for beams of elliptical cross-section. J. Sound Vib. 100(1), 1–6 (1985)
M. Elwany, A. Barr, Some optimization problems in torsional vibration. J. Sound Vib. 57(1), 1–33 (1978)
P. Muller, Torsional-flexural waves in thin-walled open beams. J. Sound Vib. 87(1), 115–141 (1983)
C. Lim, G. Zhang, J. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)
S.S. Rao, Vibration of Continuous Systems, vol. 464 (Wiley, Hoboken, 2007)
J. He, C.M. Lilley, Surface effect on the elastic behavior of static bending nanowires. Nano Lett. 8(7), 1798–1802 (2008)
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Khosravi, F., Hosseini, S.A. & Hamidi, B.A. Torsional Vibration of nanowire with equilateral triangle cross section based on nonlocal strain gradient for various boundary conditions: comparison with hollow elliptical cross section. Eur. Phys. J. Plus 135, 318 (2020). https://doi.org/10.1140/epjp/s13360-020-00312-z
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DOI: https://doi.org/10.1140/epjp/s13360-020-00312-z