Applied Mathematics and Mechanics >
Frequency equations of nonlocal elastic micro/nanobeams with the consideration of the surface effects
Received date: 2017-12-13
Revised date: 2018-02-09
Online published: 2018-08-01
A nonlocal elastic micro/nanobeam is theoretically modeled with the consideration of the surface elasticity, the residual surface stress, and the rotatory inertia, in which the nonlocal and surface effects are considered. Three types of boundary conditions, i.e., hinged-hinged, clamped-clamped, and clamped-hinged ends, are examined. For a hinged-hinged beam, an exact and explicit natural frequency equation is derived based on the established mathematical model. The Fredholm integral equation is adopted to deduce the approximate fundamental frequency equations for the clamped-clamped and clamped-hinged beams. In sum, the explicit frequency equations for the micro/nanobeam under three types of boundary conditions are proposed to reveal the dependence of the natural frequency on the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia, providing a more convenient means in comparison with numerical computations.
H. S. ZHAO, Y. ZHANG, S. T. LIE . Frequency equations of nonlocal elastic micro/nanobeams with the consideration of the surface effects[J]. Applied Mathematics and Mechanics, 2018 , 39(8) : 1089 -1102 . DOI: 10.1007/s10483-018-2358-6
[1] TIMOSHENKO, S. P. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, 41, 744-746(1921)
[2] TIMOSHENKO, S. P. On the transverse vibrations of bars of uniform cross-section. Philosophical Magazine, 43, 125-131(1922)
[3] ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703-4710(1983)
[4] PEDDIESON, J., BUCHANAN, G. R., and MCNITT, R. P. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41, 305-312(2003)
[5] WANG, L. F. and HU, H. Y. Flexural wave propagation in single-walled carbon nanotubes. Physical Review B, 71, 195412(2005)
[6] MURMU, T. and ADHIKARI, S. Nonlocal vibration of carbon nanotubes with attached buckyballs at tip. Mechanics Research Communications, 38, 62-67(2011)
[7] SUDAK, L. J. Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. Journal of Applied Physics, 94, 7281-7287(2003)
[8] WANG, C. M., ZHANG, Y. Y., RAMESH, S. S., and KITIPORNCHAI, S. Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory. Journal of Physics D:Applied Physics, 39, 3904-3909(2006)
[9] PRADHAN, S. C. Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory. Physics Letters A, 373, 4182-4188(2009)
[10] ZHANG, Z., CHALLAMEL, N., and WANG, C. M. Eringen's small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model. Journal of Applied Physics, 114, 114902(2013)
[11] LU, P., LEE, H. P., LU, C., and ZHANG, P. Q. Dynamic properties of flexural beams using a nonlocal elasticity model. Journal of Applied Physics, 99, 073510(2006)
[12] XU, M. Free transverse vibrations of nano-to-micron scale beams. Proceedings of the Royal Society A:Mathematical, Physical and Engineering Sciences, 462, 2977-2995(2006)
[13] FANG, B., ZHEN, Y. X., ZHANG, C. P., and TANG, Y. Nonlinear vibration analysis of doublewalled carbon nanotubes based on nonlocal elasticity theory. Applied Mathematical Modelling, 37, 1096-1107(2013)
[14] KIANI, K. Vibration behavior of simply supported inclined single-walled carbon nanotubes conveying viscous fluids flow using nonlocal Rayleigh beam model. Applied Mathematical Modelling, 37, 1836-1850(2013)
[15] LEI, Y., ADHIKARI, S., and FRISWELL, M. I. Vibration of nonlocal Kelvin-Voigt viscoelastic damped Timoshenko beams. International Journal of Engineering Science, 66-67, 1-13(2013)
[16] ILKHANI, M. R., BAHRAMI, A., and HOSSEINI-HASHEMI, S. H. Free vibrations of thin rectangular nano-plates using wave propagation approach. Applied Mathematical Modelling, 40, 1287-1299(2016)
[17] ZHANG, Y. and ZHAO, Y. P. Measuring the nonlocal effects of a micro/nanobeam by the shifts of resonant frequencies. International Journal of Solids and Structures, 102-103, 259-266(2016)
[18] WANG, G. F. and FENG, X. Q. Timoshenko beam model for buckling and vibration of nanowires with surface effects. Journal of Physics D:Applied Physics, 42, 155411(2009)
[19] LI, Y., SONG, J., FANG, B., and ZHANG, J. Surface effects on the postbuckling of nanowires. Journal of Physics D:Applied Physics, 44, 425304(2011)
[20] LU, P., LEE, H. P., LU, C., and O'SHEA, S. J. Surface stress effects on the resonance properties of cantilever sensors. Physical Review B, 72, 085405(2005)
[21] WANG, G. F. and FENG, X. Q. Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Applied Physics Letters, 90, 231904(2007)
[22] HE, J. and LILLEY, C. M. Surface stress effect on bending resonance of nanowires with different boundary conditions. Applied Physics Letters, 93, 263108(2008)
[23] GURTIN, M. E., MARKENSCOFF, X., and THURSTON, R. N. Effect of surface stress on the natural frequency of thin crystals. Applied Physics Letters, 29, 529-530(1976)
[24] GURTIN, M. E., WEISSMÜLLER, J., and LARCHÉ F. A general theory of curved deformable interfaces in solids at equilibrium. Philosophical Magazine A, 78, 1093-1109(1998)
[25] FARSHI, B., ASSADI, A., and ALINIA-ZIAZI, A. Frequency analysis of nanotubes with consideration of surface effects. Applied Physics Letters, 96, 093105(2010)
[26] KOOCHI, A., HOSSEINI-TOUDESHKY, H., and ABADYAN, M. Nonlinear beam formulation incorporating surface energy and size effect:application in nano-bridges. Applied Mathematics and Mechanics, 37, 583-600(2016)
[27] SHI, W. C., LI, X. F., and LEE, K. Y. Transverse vibration of free-free beams carrying two unequal end masses. International Journal of Mechanical Sciences, 90, 251-257(2015)
[28] SHI, W. C., SHEN, Z. B., PENG, X. L., and LI, X. F. Frequency equation and resonant frequencies of free-free Timoshenko beams with unequal end masses. International Journal of Mechanical Sciences, 115-116, 406-415(2016)
[29] ZHANG, Y. Frequency spectra of nonlocal Timoshenko beams and an effective method of determining nonlocal effect. International Journal of Mechanical Sciences, 128-129, 572-582(2017)
[30] HUANG, Y. and LI, X. F. A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. Journal of Sound and Vibration, 329, 2291-2303(2010)
[31] KARLI?i?, D., KOZI?, P., and PAVLOVI?, R. Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium. Applied Mathematical Modelling, 40, 1599-1614(2016)
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