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Surface effect on dynamic stability of microcantilevers on an elastic foundation under a subtangential follower force

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Abstract

For micro/nano structures, surface elasticity, surface stress and surface mass strongly affect mechanical behaviors of 1D beam-columns. This article studies dynamic stability of microcantilevers on an elastic foundation or embedded in an elastic matrix when subjected to a subtangential follower force, where the surface effects are taken into account. An exact characteristic equation is derived for clamped–free end supports. For differential tangency coefficients, the force–frequency interaction diagram is displayed and the influences of surface elasticity modulus, residual surface tension, surface mass and the elastic foundation are analyzed for conservative and non-conservative compressive forces. When the tangency coefficient vanishes, a cantilever column subjected to a conservative tip force is reduced, and conventional Euler buckling for a compressive axial load is recovered. When the tangency coefficient does not vanish, a generalized Beck’s column with the surface effects is tackled. When the tangency coefficient exceeds certain critical value, flutter instability take places. For a fixed frequency, the critical divergency and flutter loads as a function of the tangency coefficient are given for various surface influences from residual surface tension, surface elasticity, surface mass and the stiffness of the elastic foundation. The boundary map of stability, divergence and flutter domain is shown.

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References

  • Agwa, M.A., Eltaher, M.A.: Vibration of a carbyne nanomechanical mass sensor with surface effect. Appl. Phys. A 122, 335 (2016)

    Article  Google Scholar 

  • Ansari, R., Mohammadi, V., Shojaei, M.F., Gholami, R., Rouhi, H.: Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory. Eur. J. Mech. A Solids 45, 143–152 (2014)

    Article  MathSciNet  Google Scholar 

  • Attia, M.A., Mahmoud, F.F.: Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories. Int. J. Mech. Sci. 105, 126–134 (2016)

    Article  Google Scholar 

  • Chen, T.Y., Chiu, M.S.: Effects of higher-order interface stresses on the elastic states of two-dimensional composites. Mech. Mater. 43, 212–221 (2011)

    Article  Google Scholar 

  • Chen, X., Meguid, S.A.: Asymmetric bifurcation of initially curved nanobeam. J. Appl. Mech. 82, 091003 (2015a)

    Article  Google Scholar 

  • Chen, X., Meguid, S.A.: On the parameters which govern the symmetric snap-through buckling behavior of an initially curved microbeam. Int. J. Solids Struct. 66, 77–87 (2015b)

    Article  Google Scholar 

  • Chen, X., Meguid, S.A.: Snap-through buckling of initially curved microbeam subject to an electrostatic force. Proc. R. Soc. A 471, 20150072 (2015c)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, X., Meguid, S.A.: Asymmetric bifurcation of thermally and electrically actuated functionally graded material microbeam. Proc. R. Soc. A 472, 20150597 (2016)

    Article  Google Scholar 

  • Chiu, M.S., Chen, T.Y.: Effects of high-order surface stress on buckling and resonance behavior of nanowires. Acta Mech. 223, 1473–1484 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Chiu, M.S., Chen, T.Y.: Bending and resonance behavior of nanowires based on Timoshenko beam theory with high-order surface stress effects. Phys. E 54, 149–156 (2013)

    Article  Google Scholar 

  • Cuenot, S., Fretigny, C., Demoustier-Champagne, S., Nysten, B.: Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B 69, 165410 (2004)

    Article  Google Scholar 

  • Dingreville, R., Qu, J., Cherkaoui, M.: Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J. Mech. Phys. Solids. 53, 1827–1854 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Dorignac, J., Kalinowski, A., Erramilli, S., Mohanty, P.: Dynamical response of nanomechanical oscillators in immiscible viscous fluid for in vitro biomolecular recognition. Phys. Rev. Lett. 96, 186105 (2006)

    Article  Google Scholar 

  • Elishakoff, I., Soret, C.: A consistent set of nonlocal Bresse–Timoshenko equations for nanobeams with surface effects. J. Appl. Mech. 80, 061001 (2013)

    Article  Google Scholar 

  • Farshi, B., Assadi, A., Alinia-ziazi, A.: Frequency analysis of nanotubes with consideration of surface effects. Appl. Phys. Lett. 96, 093105 (2010)

    Article  Google Scholar 

  • Gao, X.L.: A new Timoshenko beam model incorporating microstructure and surface energy effects. Acta Mech. 226, 457–474 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Gavan, K.B., Westra, H.J.R., van der Drift, E.W.J.M., Venstra, W.J., van der Zant, H.S.J.: Size-dependent effective Young’s modulus of silicon nitride cantilevers. Appl. Phys. Lett. 94, 233108 (2009)

    Article  Google Scholar 

  • Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

    Article  MATH  Google Scholar 

  • He, J., Lilley, C.M.: Surface stress effect on bending resonance of nanowires with different boundary conditions. Appl. Phys. Lett. 93, 263108 (2008a)

    Article  Google Scholar 

  • He, J., Lilley, C.M.: Surface effect on the elastic behavior of static bending nanowires. Nano Lett. 8, 1798–1802 (2008b)

    Article  Google Scholar 

  • He, Q.L., Lilley, C.M.: Resonant frequency analysis of Timoshenko nanowires with surface stress for different boundary conditions. J. Appl. Phys. 112, 074322 (2012)

    Article  Google Scholar 

  • Hosseini-Hashemi, S., Nazemnezhad, R., Rokni, H.: Nonlocal nonlinear free vibration of nanobeams with surface effects. Eur. J. Mech. A. Solids. 52, 44–53 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Kiani, K.: Forced vibrations of a current-carrying nanowire in a longitudinal magnetic field accounting for both surface energy and size effects. Phys. E. 63, 27–35 (2014)

    Article  Google Scholar 

  • Kiani, K.: Axial buckling analysis of a slender current-carrying nanowire acted upon by a magnetic field using the surface energy approach. J. Phys. D Appl. Phys. 48, 245302 (2015)

    Article  Google Scholar 

  • Lachut, M.J., Sader, J.E.: Effect of surface stress on the stiffness of cantilever plates. Phys. Rev. Lett. 99, 206102 (2007)

    Article  Google Scholar 

  • Lee, H.L., Chang, W.J.: Surface effects on frequency analysis of nanotubes using nonlocal Timoshenko beam theory. J. Appl. Phys. 108, 093503 (2010)

    Article  Google Scholar 

  • Li, X.-F., Peng, X.-L.: Theoretical analysis of surface stress for a microcantilever with varying widths. J. Phys. D Appl. Phys. 41, 065301 (2008)

    Article  Google Scholar 

  • Li, X.-F., Zhang, H., Lee, K.Y.: Dependence of Young’s modulus of nanowires on surface effect. Int. J. Mech. Sci. 81, 120–125 (2014)

    Article  Google Scholar 

  • Li, X.F., Zou, J., Jiang, S.N., Lee, K.Y.: Resonant frequency and flutter instability of a nanocantilever with the surface effects. Compos. Struct. 153, 645–653 (2016)

    Article  Google Scholar 

  • Liu, C., Rajapakse, R.: Continuum models incorporating surface energy for static and dynamic response of nanoscale beams. IEEE Trans. Nanotechnol. 9, 422–431 (2010)

    Article  Google Scholar 

  • Lu, P., He, L.H., Lee, H.P., Lu, C.: Thin plate theory including surface effects. Int. J. Solids Struct. 43, 4631–4647 (2006)

    Article  MATH  Google Scholar 

  • McFarland, A.W., Poggi, M.A., Doyle, M.J., Bottomley, L.A., Colton, J.S.: Influence of surface stress on the resonance behavior of microcantilevers. Appl. Phys. Lett. 87, 53505–53505 (2005)

    Article  Google Scholar 

  • Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139 (2000)

    Article  Google Scholar 

  • Nazemnezhad, R., Hosseini-Hashemi, S.: Nonlinear free vibration analysis of Timoshenko nanobeams with surface energy. Meccanica 50, 1027–1044 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Qiao, L., Zheng, X.J.: Effect of surface stress on the stiffness of micro/nanocantilevers: nanowire elastic modulus measured by nano-scale tensile and vibrational techniques. J. Appl. Phys. 113, 013508 (2013)

    Article  Google Scholar 

  • Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B 71, 094104 (2005)

    Article  Google Scholar 

  • Simitses, G.J., Hodges, D.H.: Fundamentals of Structural Stability. Butterworth-Heinemann, London (2006)

    MATH  Google Scholar 

  • Wang, L.: Vibration analysis of fluid-conveying nanotubes with consideration of surface effects. Phys. E. 43, 437–439 (2010)

    Article  Google Scholar 

  • Wang, G.-F., Feng, X.-Q.: Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl. Phys. Lett. 90, 231904 (2007)

    Article  Google Scholar 

  • Wang, G.-F., Feng, X.-Q.: Timoshenko beam model for buckling and vibration of nanowires with surface effects. J. Phys. D Appl. Phys. 42, 155411 (2009)

    Article  Google Scholar 

  • Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., Wang, T.: Surface stress effect in mechanics of nanostructured materials. Acta Mech. Solid Sin. 24, 52–82 (2011)

    Article  Google Scholar 

  • Wang, H., Li, X., Tang, G., Shen, Z.: Effect of surface stress on stress intensity factors of a nanoscale crack via double cantilever beam model. J. Nanosci. Nanotechnol. 13, 477–482 (2013)

    Article  Google Scholar 

  • Wang, K.F., Wang, B.L.: A general model for nano-cantilever switches with consideration of surface effects and nonlinear curvature. Phys. E. 66, 197–208 (2015)

    Article  Google Scholar 

  • Weaver Jr., W., Timoshenko, S.P., Young, D.H.: Vibration Problems in Engineering. Wiley, New York (1990)

    Google Scholar 

  • Wu, J.-X., Li, X.-F., Tang, A.-Y., Lee, K.Y.: Free and forced transverse vibration of nanowires with surface effects. J. Vib. Control. 1077546315610302 (2016)

  • Yi, X., Duan, H.L.: Surface stress induced by interactions of adsorbates and its effect on deformation and frequency of microcantilever sensors. J. Mech. Phys. Solids. 57, 1254–1266 (2009)

    Article  MATH  Google Scholar 

  • Zhang, J., Meguid, S.A.: Effect of surface energy on the dynamic response and instability of fluid-conveying nanobeams. Eur. J. Mech. A Solids. 58, 1–9 (2016)

    Article  MathSciNet  Google Scholar 

  • Zhang, Y.Q., Pang, M., Chen, W.Q.: Transverse vibrations of embedded nanowires under axial compression with high-order surface stress effects. Phys. E. 66, 238–244 (2015)

    Article  Google Scholar 

  • Zhang, Y., Ren, Q., Zhao, Y.-P.: Modelling analysis of surface stress on a rectangular cantilever beam. J. Phys. D Appl Phys. 37, 2140 (2004)

    Article  Google Scholar 

  • Zheng, X.P., Cao, Y.P., Li, B., Feng, X.Q., Wang, G.F.: Surface effects in various bending-based test methods for measuring the elastic property of nanowires. Nanotechnology 21, 205702 (2010)

    Article  Google Scholar 

  • Zuo, Q.H., Schreyer, H.L.: Flutter and divergence instability of nonconservative beams and plates. Int. J. Solids Struct. 33, 1355–1367 (1996)

    Article  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11672336) and the Open Foundation of State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, PRC (No. GZ15204).

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Authors and Affiliations

  1. School of Civil Engineering, Central South University, Changsha, 410075, People’s Republic of China

    X.-F. Li & S.-N. Jiang

  2. State Key Laboratory of Structural Analysis for Industrial Equipment and Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    X.-F. Li & K. Y. Lee

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  1. X.-F. Li
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  2. S.-N. Jiang
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Correspondence to K. Y. Lee.

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Li, XF., Jiang, SN. & Lee, K.Y. Surface effect on dynamic stability of microcantilevers on an elastic foundation under a subtangential follower force. Int J Mech Mater Des 14, 91–104 (2018). https://doi.org/10.1007/s10999-016-9362-1

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  • DOI: https://doi.org/10.1007/s10999-016-9362-1

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