Size-Dependent Forced Vibration Analysis of Three Nonlocal Strain Gradient Beam Models with Surface Effects Subjected to Moving Harmonic Loads

Subject Areas : Mechanical Engineering

K Rajabi ¹ , Sh Hosseini Hashemi ² , A.R Nezamabadi ³

1 - Department of Mechanical Engineering, College of Engineering, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran
2 - School of Mechanical Engineering , Iran University of Science and Technology, Tehran, Iran
3 - Department of Mechanical Engineering, Arak Branch, Islamic Azad University, Arak, Iran

Received: 2018-09-06 Accepted : 2018-12-08 Published : 2019-03-30

Keywords: Timoshenko beam model, Analytical solution, Moving harmonic load, Euler-Bernoulli beam model, Nonlocal strain gradient elasticity theory,

Abstract :

The forced vibration behaviors are examined for nonlocal strain gradient nanobeams with surface effects subjected to a moving harmonic load travelling with a constant velocity in terms of three beam models namely, the Euler-Bernoulli, Timoshenko and modified Timoshenko beam models. The modification for nonlocal strain gradient Timoshenko nanobeams is exerted to the constitutive equations by exclusion of the nonlocality in the shear constitutive relation. Some analytical closed-form solutions for three nonlocal strain gradient beam models with simply supported boundary conditions are derived by using the Galerkin discretization method in conjunction with the Laplace transform method. The effects of the three beam models, the nonlocal and material length scale parameters, the velocity and excitation frequency of the moving harmonic load on the dynamic behaviors of nanobeams are discussed in some detail. Specifically, the critical velocities are examined in some detail. Numerical results have shown that the aforementioned parameters are very important factors for determining the dynamic behavior of the nanobeams accurately.

References:

[1] Kiani K., 2010, Longitudinal an7d transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effects, Physica E: Low-dimensional Systems and Nanostructures 42(9): 2391-2401.
[2] Pijper D., 2005, Acceleration of a nanomotor: electronic control of the rotary speed of a light-driven molecular rotor, Journal of the American Chemical Society 127(50): 17612-17613.
[3] Shirai Y., 2005, Directional control in thermally driven single-molecule nanocars, Nano Letters 5(11): 2330-2334.
[4] Shirai Y., 2006, Surface-rolling molecules, Journal of the American Chemical Society 128(14): 4854-4864.
[5] Shirai Y., 2006, Recent progress on nanovehicles, Chemical Society Reviews 35(11): 1043-1055.
[6] Gross L., 2005, Trapping and moving metal atoms with a six-leg molecule, Nature Materials 4(12): 892-895.
[7] Kiani K., Mehri B., 2010, Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories, Journal of Sound and Vibration 329(11): 2241-2264.
[8] Kiani K., 2010, Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle, Part I: Theoretical formulations, Acta Mechanica 216(1-4): 165-195.
[9] Kiani K., 2011, Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle, Part I: Theoretical formulations, Physica E: Low-dimensional Systems and Nanostructures 44(1): 229-248.
[10] Kiani K., Wang Q., 2012, On the interaction of a single-walled carbon nanotube with a moving nanoparticle using nonlocal Rayleigh, Timoshenko, and higher-order beam theories, European Journal of Mechanics - A/Solids 31(1): 179-202.
[11] Kiani K., 2010, Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle, Part II: Parametric study, Acta Mechanica 216(1-4): 197-206.
[12] Arani A.G., Roudbari M., Amir S., 2012, Nonlocal vibration of SWBNNT embedded in bundle of CNTs under a moving nanoparticle, Physica B: Condensed Matter 407(17): 3646-3653.
[13] Kiani K., 2011, Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle, Part II: Parametric studies, Physica E: Low-dimensional Systems and Nanostructures 44(1): 249-269.
[14] Picu C., 2003, A nonlocal formulation of rubber elasticity, International Journal for Multiscale Computational Engineering 1(1): 23-32.
[15] Şimşek M., 2010, Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory, Physica E: Low-dimensional Systems and Nanostructures 43(1): 182-191.
[16] Bazant Z.P., Jirásek M., 2002, Nonlocal integral formulations of plasticity and damage: survey of progress, Journal of Engineering Mechanics 128(11): 1119-1149.
[17] Jirasek M., 2004, Nonlocal theories in continuum mechanics, Acta Polytechnica 44(5-6): 16-34.
[18] Yi D., Wang T.C., Xiao Z., 2010, Strain gradient theory based on a new framework of non-local model, Acta Mechanica 212(1-2): 51-67.
[19] Akgöz B., Civalek Ö., 2011, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science 49(11): 1268-1280.
[20] Akgöz B., Civalek Ö., 2012, Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory, Archive of Applied Mechanics 82(3): 423-443.
[21] Wu J., Li X., Cao W., 2013, Flexural waves in multi-walled carbon nanotubes using gradient elasticity beam theory, Computational Materials Science 67: 188-195.
[22] Peddieson J., Buchanan G.R., McNitt R.P., 2003, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 41(3): 305-312.
[23] Lu P., 2006, Dynamic properties of flexural beams using a nonlocal elasticity model, Journal of Applied Physics 99(7): 073510.
[24] Reddy J., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science 45(2): 288-307.
[25] Murmu T., Pradhan S., 2009, Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Computational Materials Science 46(4): 854-859.
[26] Askes H., Aifantis E.C., 2011, Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results, International Journal of Solids and Structures 48(13): 1962-1990.
[27] Tian Y., 2013, Ultrahard nanotwinned cubic boron nitride, Nature 493(7432): 385-388.
[28] Li X., 2010, Dislocation nucleation governed softening and maximum strength in nano-twinned metals, Nature 464(7290): 877-880.
[29] Lim C., Zhang G., Reddy J., 2015, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids 78: 298-313.
[30] Ebrahimi F., Barati M.R., 2016, Flexural wave propagation analysis of embedded S-FGM nanobeams under longitudinal magnetic field based on nonlocal strain gradient theory, Arabian Journal for Science and Engineering 2016: 1-12.
[31] Farajpour A., 2016, A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment, Acta Mechanica 2016: 1-19.
[32] Hosseini S., Rahmani O., 2016, Exact solution for axial and transverse dynamic response of functionally graded nanobeam under moving constant load based on nonlocal elasticity theory, Meccanica 2016: 1-17.
[33] Li L., Hu Y., 2016, Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory, Computational Materials Science 112: 282-288.
[34] Li L., Hu Y., Li X., 2016, Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory, International Journal of Mechanical Sciences 115: 135-144.
[35] Li L., 2016, Size-dependent effects on critical flow velocity of fluid-conveying microtubes via nonlocal strain gradient theory, Microfluidics and Nanofluidics 20(5): 1-12.
[36] Li L., Li X., Hu Y., 2016, Free vibration analysis of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science 102: 77-92.
[37] Şimşek M., 2016, Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science 105: 12-27.
[38] Şimşek M., 2016, Axial vibration analysis of a nanorod embedded in elastic medium using nonlocal strain gradient theory, Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi 31(1): 213-221.
[39] Tang Y., Liu Y., Zhao D., 2016, Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory, Physica E: Low-dimensional Systems and Nanostructures 84: 202-208.
[40] Fernandes R., 2017, Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium, Physica E: Low-dimensional Systems and Nanostructures 88: 18-25.
[41] Shen Y., Chen Y., Li L., 2016, Torsion of a functionally graded material, International Journal of Engineering Science 109: 14-28.
[42] Guo S., 2016, Torsional vibration of carbon nanotube with axial velocity and velocity gradient effect, International Journal of Mechanical Sciences 119: 88-96.
[43] Li X., 2017, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures 165: 250-265.
[44] Li L., Hu Y., 2017, Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects, International Journal of Mechanical Sciences 120: 159-170.
[45] Ebrahimi F., Barati M.R., Dabbagh A., 2016, A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates, International Journal of Engineering Science 107: 169-182.
[46] Wang G.-F., Feng X.-Q., 2009, Surface effects on buckling of nanowires under uniaxial compression, Applied Physics Letters 94(14): 141913.
[47] Hosseini S.A.H., Rahmani O., 2016, Surface effects on buckling of double nanobeam system based on nonlocal timoshenko model, International Journal of Structural Stability and Dynamics 16(10): 1550077.
[48] He J., Lilley C.M., 2008, Surface effect on the elastic behavior of static bending nanowires, Nano Letters 8(7): 1798-1802.
[49] Farshi B., Assadi A., Alinia-ziazi A., 2010, Frequency analysis of nanotubes with consideration of surface effects, Applied Physics Letters 96(9): 093105.
[50] Abbasion S., 2009, Free vibration of microscaled Timoshenko beams, Applied Physics Letters 95(14): 143122.
[51] Ansari R., 2014, Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory, European Journal of Mechanics - A/Solids 45: 143-152.
[52] Lei X.-w., 2012, Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model, Composites Part B: Engineering 43(1): 64-69.
[53] Lee H.-L., Chang W.-J., 2010, Surface effects on frequency analysis of nanotubes using nonlocal Timoshenko beam theory, Journal of Applied Physics 108(9): 093503.
[54] Wang G.-F., Feng X.-Q., 2009, Timoshenko beam model for buckling and vibration of nanowires with surface effects, Journal of Physics D: Applied Physics 42(15): 155411.
[55] Arash B., Wang Q., 2012, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Computational Materials Science 51(1): 303-313.
[56] Wang Q., Wang C.M., 2007, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology 18(7): 075702.
[57] Younesian D., Nankali A., Motieyan E., 2011, Optimal nonlinear energy sinks in vibration mitigation of the beams traversed by successive moving loads, Journal of Solid Mechanics 3(4): 323-331.
[58] Rajabi K., Kargarnovin M., Gharini M., 2013, Dynamic analysis of a functionally graded simply supported Euler–Bernoulli beam subjected to a moving oscillator, Acta Mechanica 2013: 1-22.
[59] Pang M., Zhang Y.Q., Chen W.Q., 2015, Transverse wave propagation in viscoelastic single-walled carbon nanotubes with small scale and surface effects, Journal of Applied Physics 117(2): 024305.
[60] Naderi A., Saidi A., 2013, Modified nonlocal mindlin plate theory for buckling analysis of nanoplates, Journal of Nanomechanics and Micromechanics 4(4): A4013015.
[61] Shenoy V.B., 2005, Atomistic calculations of elastic properties of metallic fcc crystal surfaces, Physical Review B 71(9): 094104.