Structural Engineering and Mechanics

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Size-dependent thermal behaviors of axially traveling nanobeams based on a strain gradient theory

  • Li, Cheng (School of Urban Rail Transportation, Soochow University)
  • Received : 2013.08.22
  • Accepted : 2013.10.26
  • Published : 2013.11.10
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Abstract

This work is concerned with transverse vibrations of axially traveling nanobeams including strain gradient and thermal effects. The strain gradient elasticity theory and the temperature field are taken into consideration. A new higher-order differential equation of motion is derived from the variational principle and the corresponding higher-order non-classical boundary conditions including simple, clamped, cantilevered supports and their higher-order "offspring" are established. Effects of strain gradient nanoscale parameter, temperature change, shape parameter and axial traction on the natural frequencies are presented and discussed through some numerical examples. It is concluded that the factors mentioned above significantly influence the dynamic behaviors of an axially traveling nanobeam. In particular, the strain gradient effect tends to induce higher vibration frequencies as compared to an axially traveling macro beams based on the classical vibration theory without strain gradient effect.

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References

  1. Aksencer, T. and Aydogdu, M. (2012), "Forced transverse vibration of nanoplates using nonlocal elasticity", Phys. E, 44, 1752-1759. https://doi.org/10.1016/j.physe.2011.12.004
  2. Avsec, J. and Oblak, M. (2007), "Thermal vibrational analysis for simply supported beam and clamped beam", J. Sound Vib., 308, 514-525. https://doi.org/10.1016/j.jsv.2007.04.002
  3. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  4. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803
  5. Eringen, A.C., Speziale, C.G. and Kim, B.S. (1977), "Crack tip problem in non-local elasticity", J. Mech. Phys. Solids, 25, 339-355. https://doi.org/10.1016/0022-5096(77)90002-3
  6. Hatami, S., Azhari, M. and Saadatpour, M.M. (2007), "Free vibration of moving laminated composite plates", Compos. Struct., 80, 609-620. https://doi.org/10.1016/j.compstruct.2006.07.009
  7. Lee, H.L. and Chang, W.J. (2009), "Vibration analysis of a viscous-fluid-conveying single-walled carbon nanotube embedded in an elastic medium", Phys. E, 41, 529-532. https://doi.org/10.1016/j.physe.2008.10.002
  8. Li, C., Lim, C.W. and Yu, J.L. (2011a), "Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load", Smart Mater. Struct., 20, 015023. https://doi.org/10.1088/0964-1726/20/1/015023
  9. Li, C., Lim, C.W., Yu, J.L. and Zeng, Q.C. (2011b), "Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force", Int. J. Struct. Stab. Dyn., 11, 257-271. https://doi.org/10.1142/S0219455411004087
  10. Li, C., Lim, C.W., Yu, J.L. and Zeng, Q.C. (2011c), "Transverse vibration of pre-tensioned nonlocal nanobeams with precise internal axial loads", Sci. Chin. Tech. Sci., 54, 2007-2013. https://doi.org/10.1007/s11431-011-4479-9
  11. Li, C., Zheng, Z.J., Yu, J.L. and Lim, C.W. (2011d), "Static analysis of ultra-thin beams based on a semicontinuum moder", Acta Mech. Sin., 27, 713-719. https://doi.org/10.1007/s10409-011-0453-9
  12. Lim, C.W. (2010), "On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection", Appl. Math. Mech., 31, 37-54. https://doi.org/10.1007/s10483-010-0105-7
  13. Lim, C.W., Li. C. and Yu, J.L. (2009), "The effects of stiffness strengthening nonlocal stress and axial tension on free vibration of cantilever nanobeams", Interaction Multiscale Mech., 2, 223-233. https://doi.org/10.12989/imm.2009.2.3.223
  14. Lim, C.W., Li. C. and Yu, J.L. (2012), "Free torsional vibration of nanotubes based on nonlocal stress theory", J. Sound Vib., 331, 2798-2808. https://doi.org/10.1016/j.jsv.2012.01.016
  15. Lim, C.W., Niu, J.C. and Yu, Y.M. (2010), "Nonlocal stress theory for buckling instability of nanotubes: new predictions on stiffness strengthening effects of nanoscales", J. Comput. Theore. Nanosci., 7, 2104-2111. https://doi.org/10.1166/jctn.2010.1591
  16. Lim, C.W. and Yang, Q. (2011), "Nonlocal thermal-elasticity for nanobeam deformation: exact solutions with stiffness enhancement effects", J. Appl. Phys., 110, 013514. https://doi.org/10.1063/1.3596568
  17. Lu, P., Lee, H.P., Lu, C. and Zhang, P.Q. (2006), "Dynamic properties of flexural beams using a nonlocal elasticity model", J. Appl. Phys., 99, 073510. https://doi.org/10.1063/1.2189213
  18. Mindlin, R.D. (1965), "Second gradient of strain and surface tension in linear elasticity", Int. J. Solids Struct., 1, 417-438. https://doi.org/10.1016/0020-7683(65)90006-5
  19. Oz, H.R. and Pakdemirli, M. (1999), "Vibrations of an axially moving beam with time-dependent velocity", J. Sound Vib., 227, 239-257. https://doi.org/10.1006/jsvi.1999.2247
  20. Pakdemirli, M. and Oz. H.R. (2008), "Infinite mode analysis and truncation to resonant modes of axially accelerating beam vibrations", J. Sound Vib., 311, 1052-1074. https://doi.org/10.1016/j.jsv.2007.10.003
  21. Peddieson, J., Buchanan, G.G. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41, 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
  22. Pradhan, S.C. and Kumar, A. (2011), "Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method", Compos. Stroct., 93, 774-779. https://doi.org/10.1016/j.compstruct.2010.08.004
  23. Tang, Y.Q., Chen, L.Q. and Yang, X.D. (2009), "Nonlinear vibrations of axially moving Timoshenko beams under weak and strong external excitations", J. Sound Vib., 320, 1078-1099. https://doi.org/10.1016/j.jsv.2008.08.024
  24. Wang, L. (2011), "A modified nonlocal beam model for vibration and stability of nanotubes conveying fluid", Phys. E, 44, 25-28. https://doi.org/10.1016/j.physe.2011.06.031
  25. Wang, Q., Zhou, G.Y. and Lin, K.C. (2006), "Scale effect on wave propagation of double-walled carbon nanotubes", Int. J. Solids Struct, 43, 6071-6084. https://doi.org/10.1016/j.ijsolstr.2005.11.005
  26. Yao, X.H. and Han, Q. (2007), "Investigation of axially compressed buckling of a multi-walled carbon nanotube under temperature field", Compos. Sci. Tech., 67, 125-134. https://doi.org/10.1016/j.compscitech.2006.03.021
  27. Yu, Y.M. and Lim, C.W. (2013), "Nonlinear constitutive model for axisymmetric bending of annular graphene-like nanoplate with gradient elasticity enhancement effects", J. Eng. Mech.-ASCE, 139, 1025-1035. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000625
  28. Zhang, Y.Q., Liu, G.R. and Wang, J.S. (2004), "Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression", Phys. Rev. B, 70, 205430. https://doi.org/10.1103/PhysRevB.70.205430

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