An Analytical Study on Surface Energy Effect on Free Longitudinal Vibration of Cracked Nanorods
Subject Areas : Applied Mechanics
Hassan Shokrollahi
1
,
Reza Nazemnezhad
2
1 - Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran
2 - School of Engineering, Damghan University, Damghan, Iran
Keywords: Surface energy, Cracked nanorod, longitudinal vibration,
Abstract :
The present work analytically studies the free longitudinal vibration of nanorods in the presence of cracks based on the surface elasticity theory. To this end, governing equations of motion and corresponding boundary conditions are obtained using Hamilton’s principle. Due to considering the surface stress effect, as well as the surface density and the surface Lamé constants, the obtained governing equations of motion become non-homogeneous. The non-homogeneous governing equations are solved using appropriate analytical methods, and the natural frequencies are extracted. To have a comprehensive research, the effects of various parameters such as the length and radius of the nanorod, the crack severity, the crack position, the type of boundary condition, and the values of surface and bulk material properties on axial frequencies of the nanorod are investigated. Since this work considers the effects of all surface energy parameters, it can be claimed that it is a comprehensive study in this regard.
[1] Nowak A.S., Collins K.R., 2012, Reliability of structures, CRC Press.
[2] Grassian V.H., 2008, Nanoscience and Nanotechnology: Environmental and Health Impacts, Wiley.
[3] Klabunde K.J., 2001, Introduction to Nanotechnology, In Nanoscale Materials in Chemistry, https://doi.org/10.1002/0471220620.ch1.
[4] Wang M.F., Du G.J., Xia D.Y., 2013, Molecular dynamics simulation of microcrack healing in copper nano-plate, Key Engineering Materials, doi:10.4028/www.scientific.net/kem.531-532.454.
[5] Ding J., Wang L.S., Song K., Liu B., Huang X., 2017, Molecular dynamics simulation of crack propagation in single-crystal aluminum plate with central cracks, Journal of Nanomaterials, doi:10.1155/2017/5181206.
[6] Hu Z.L., Lee K., Li X.F., 2018, Crack in an elastic thin-film with surface effect, International Journal of Engineering Science 123:158-73.
[7] Loya J., López-Puente J., Zaera R., Fernández-Sáez J., 2009, Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model, Journal of Applied Physics 105:044309.
[8] Safarabadi M., Mohammadi M., Farajpour A., Goodarzi M., 2015, Effect of surface energy on the vibration analysis of rotating nanobeam, Journal of Solid Mechanics 7:299-311.
[9] Rahmanian S., Hosseini Hashemi S., 2019, Bifurcation and chaos in size-dependent nems considering surface energy effect and intermolecular interactions, Journal of Solid Mechanics 11:341-60.
[10] Demir C., Mercan K., Numanoglu H.M., Civalek O., 2018, Bending response of nanobeams resting on elastic foundation, Journal of Applied and Computational Mechanics 4:105-14.
[11] Torabi K., Dastgerdi J.N., 2012, An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model, Thin Solid Films 520:6595-602.
[12] Ghadiri M., Soltanpour M., Yazdi A., Safi M., 2016, Studying the influence of surface effects on vibration behavior of size-dependent cracked FG Timoshenko nanobeam considering nonlocal elasticity and elastic foundation, Applied Physics A 122:520.
[13] Hosseini-Hashemi S., Fakher M., Nazemnezhad R., 2013, Surface effects on free vibration analysis of nanobeams using nonlocal elasticity: a comparison between Euler-Bernoulli and Timoshenko, Journal of Solid Mechanics 5:290-304.
[14] Numanoğlu H.M., Ersoy H., Akgöz B., Civalek Ö., 2022, A new eigenvalue problem solver for thermo-mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method, Mathematical Methods in the Applied Sciences 45:2592-614.
[15] Soltanpour M., Ghadiri M., Yazdi A., Safi M., 2017, Free transverse vibration analysis of size dependent Timoshenko FG cracked nanobeams resting on elastic medium, Microsystem Technologies 23:1813-30.
[16] Jalaei M.H., Thai H.T., Civalek Ӧ., 2022, On viscoelastic transient response of magnetically imperfect functionally graded nanobeams, International Journal of Engineering Science 172:103629.
[17] Roostai H., Haghpanahi M., 2014, Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory, Applied Mathematical Modelling 38:1159-69.
[18] Karličić D., Jovanović D., Kozić P., Cajić M., 2015, Thermal and magnetic effects on the vibration of a cracked nanobeam embedded in an elastic medium, Journal of Mechanics of Materials and Structures 10:43-62.
[19] Khorshidi M.A., Shaat M., Abdelkefi A., Shariati M., 2017, Nonlocal modeling and buckling features of cracked nanobeams with von Karman nonlinearity, Applied Physics A 123:62.
[20] Hasheminejad S.M., Gheshlaghi B., Mirzaei Y., Abbasion S., 2011, Free transverse vibrations of cracked nanobeams with surface effects, Thin Solid Films 519:2477-82.
[21] Hosseini-Hashemi S., Fakher M., Nazemnezhad R., Haghighi M.H.S., 2014, Dynamic behavior of thin and thick cracked nanobeams incorporating surface effects, Composites Part B: Engineering 61:66-72.
[22] Wang K., Wang B., 2015, Timoshenko beam model for the vibration analysis of a cracked nanobeam with surface energy, Journal of Vibration and Control 21:2452-64.
[23] Hu K.M., Zhang W.M., Peng Z.K., Meng G., 2016, Transverse vibrations of mixed-mode cracked nanobeams with surface effect, Journal of Vibration and Acoustics 138:011020.
[24] Akbas S.D., 2016, Analytical solutions for static bending of edge cracked micro beams, Structural Engineering and Mechanics 59:579-99.
[25] Khorshidi M.A., Shariati M., 2017, Buckling and postbuckling of size-dependent cracked microbeams based on a modified couple stress theory, Journal of Applied Mechanics and Technical Physics 58:717-24.
[26] Khorshidi M.A., Shariati M., 2017, A multi-spring model for buckling analysis of cracked Timoshenko nanobeams based on modified couple stress theory, Journal of Theoretical and Applied Mechanics 55:1127-39.
[27] Beni Y.T., Jafaria A., Razavi H., 2014, Size effect on free transverse vibration of cracked nano-beams using couple stress theory, International Journal of Engineering-Transactions B: Applications 28:296-304.
[28] Sourki R., Hoseini S., 2016, Free vibration analysis of size-dependent cracked microbeam based on the modified couple stress theory, Applied Physics A. 122:413.
[29] Akbaş Ş.D., 2017, Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory, International Journal of Structural Stability and Dynamics 17:1750033.
[30] Loya J., Aranda-Ruiz J., Fernández-Sáez J., 2014, Torsion of cracked nanorods using a nonlocal elasticity model, Journal of Physics D: Applied Physics 47:115304.
[31] Rahmani O., Hosseini S., Noroozi Moghaddam M., Fakhari Golpayegani I., 2015, Torsional vibration of cracked nanobeam based on nonlocal stress theory with various boundary conditions: an analytical study, International Journal of Applied Mechanics 7:1550036.
[32] Nazemnezhad R., Fahimi P., 2017, Free torsional vibration of cracked nanobeams incorporating surface energy effects, Applied Mathematics and Mechanics 38:217-30.
[33] Hsu J.C., Lee H.L., Chang W.J., 2011, Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory, Current Applied Physics 11:1384-8.
[34] Yaylı M.Ö., Çerçevik A.E., 2015, Axial vibration analysis of cracked nanorods with arbitrary boundary conditions, Journal of Vibroengineering 17.
[35] Hosseini A.H., Rahmani O., Nikmehr M., Golpayegani I.F., 2016, Axial vibration of cracked nanorods embedded in elastic foundation based on a nonlocal elasticity model, Sensor Letters 14:1019-25.
[36] Rao S.S., 2007, Vibration of continuous systems, John Wiley & Sons.
[37] Gurtin M.E., Murdoch A.I., 1975, A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis 57:291-323.
[38] Nazemnezhad R., Shokrollahi H., 2019, Free axial vibration analysis of functionally graded nanorods using surface elasticity theory, Modares Mechanical Engineering 18:131-41.
[39] Nazemnezhad R., Shokrollahi H., 2020, Free axial vibration of cracked axially functionally graded nanoscale rods incorporating surface effect, Steel and Composite Structures 35:449-62.