Free torsional vibration of cracked nanobeams incorporating surface energy effects

doi:10.1007/s10483-017-2167-9

Abstract

Abstract:

This paper investigates surface energy effects, including the surface shear modulus, the surface stress, and the surface density, on the free torsional vibration of nanobeams with a circumferential crack and various boundary conditions. To formulate the problem, the surface elasticity theory is used. The cracked nanobeam is modeled by dividing it into two parts connected by a torsional linear spring in which its stiffness is related to the crack severity. Governing equations and corresponding boundary conditions are derived with the aid of Hamilton's principle. Then, natural frequencies are obtained analytically, and the influence of the crack severity and position, the surface energy, the boundary conditions, the mode number, and the dimensions of nanobeam on the free torsional vibration of nanobeams is studied in detail. Results of the present study reveal that the surface energy has completely different effects on the free torsional vibration of cracked nanobeams compared with its effects on the free transverse vibration of cracked nanobeams.

Key words: cracked nanobeam, surface energy, torsional vibration

2010 MSC Number:

O322

R. NAZEMNEZHAD, P. FAHIMI. Free torsional vibration of cracked nanobeams incorporating surface energy effects. Applied Mathematics and Mechanics (English Edition), 2017, 38(2): 217-230.

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References

[1] Fei, P., Yeh, P. H., Zhou, J., Xu, S., Gao, Y., Song, J., Gu, Y., Huang, Y., and Wang, Z. L. Piezoelectric potential gated field-effect transistor based on a free-standing ZnO wire. Nano Lett., 9(10), 3435-3439(2009)
[2] He, J. H., Hsin, C. L., Liu, J., Chen, L. J., and Wang, Z. L. Piezoelectric gated diode of a single ZnO nanowire. Adv. Mater., 19(6), 781-784(2007)
[3] Wang, Z. L. and Song, J. Piezoelectric nanogenerators based on zinc oxide nanowire arrays. Science, 312(5771), 242-246(2006)
[4] Zhong, Z., Wang, D., Cui, Y., Bockrath, M. W., and Lieber, C. M. Nanowire crossbar arrays as address decoders for integrated nanosystems. Science, 302(5649), 1377-1379(2003)
[5] Bai, X., Gao, P., Wang, Z. L., and Wang, E. Dual-mode mechanical resonance of individual ZnO nanobelts. Appl. Phys. Lett., 82(26), 4806-4808(2003)
[6] Nazemnezhad, R. and Hosseini-Hashemi, S. Nonlinear free vibration analysis of Timoshenko nanobeams with surface energy. Meccanica, 50(4), 1027-1044(2015)
[7] Hosseini-Hashemi, S., Nazemnezhad, R., and Bedroud, M. Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity. Appl. Math. Model., 38(14), 3538-3553(2014)
[8] Hosseini-Hashemi, S., Nahas, I., Fakher, M., and Nazemnezhad, R. Nonlinear free vibration of piezoelectric nanobeams incorporating surface effects. Smart. Mater. Struct., 23(3), 035012(2014)
[9] Hosseini-Hashemi, S., Nahas, I., Fakher, M., and Nazemnezhad, R. Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity. Acta Mech., 225(6), 1555-1564(2014)
[10] Liu, C. and Rajapakse, R. Continuum models incorporating surface energy for static and dynamic response of nanoscale beams. IEEE Trans. Nanotechnol., 9(4), 422-431(2010)
[11] Gurtin, M. E. and Murdoch, A. I. A continuum theory of elastic material surfaces. Arch. Ration. Mech. An., 57(4), 291-323(1975)
[12] Gurtin, M. E. and Murdoch, A. I. Surface stress in solids. Int. J. Solids. Struct., 14(6), 431-440(1978)
[13] Hosseini-Hashemi, S., Nazemnezhad, R., and Rokni, H. Nonlocal nonlinear free vibration of nanobeams with surface effects. Eur. J. Mech. A-Solids., 52, 44-53(2015)
[14] Ansari, R., Mohammadi, V., Shojaei, M. F, Gholami, R., and Sahmani, S. On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory. Compos. Part BEng, 60, 158-166(2014)
[15] Wang, G. F. and Feng, X. Q. Timoshenko beam model for buckling and vibration of nanowires with surface effects. J. Phys. D. Appl. Phys., 42(15), 155411(2009)
[16] Li, Y., Chen, C., Fang, B., Zhang, J., and Song, J. Postbuckling of piezoelectric nanobeams with surface effects. Int. J. Appl. Mech., 4(2), 1250018(2012)
[17] Guo, J. G. and Zhao, Y. P. The size-dependent bending elastic properties of nanobeams with surface effects. Nanotechnology, 18(29), 295701(2007)
[18] Assadi, A. and Farshi, B. Size-dependent longitudinal and transverse wave propagation in embedded nanotubes with consideration of surface effects. Acta Mech., 222(1/2), 27-39(2011)
[19] Hosseini-Hashemi, S., Fakher, M., Nazemnezhad, R., and Haghighi, M. H. S. Dynamic behavior of thin and thick cracked nanobeams incorporating surface effects. Compos. Part B-Eng., 61, 66-72(2014)
[20] Wang, K. and Wang, B. Timoshenko beam model for the vibration analysis of a cracked nanobeam with surface energy. J. Vib. Control., 21(12), 2452-2464(2015)
[21] Hasheminejad, S. M., Gheshlaghi, B., Mirzaei, Y., and Abbasion, S. Free transverse vibrations of cracked nanobeams with surface effects. Thin. Solid. Films, 519(8), 2477-2482(2011)
[22] Wang, G. F. and Feng, X. Q. Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl. Phys. Lett., 90(23), 231904(2007)
[23] Nazemnezhad, R., Salimi, M., Hashemi, S. H., and Sharabiani, P. A. An analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density effects. Compos. Part B-Eng., 43, 2893-2897(2012)
[24] Fennimore, A., Yuzvinsky, T., Han, W. Q., Fuhrer, M., Cumings, J., and Zettl, A. Rotational actuators based on carbon nanotubes. nature, 424(6947), 408-410(2003)
[25] Witkamp, B., Poot, M., Pathangi, H., Hüttel, A., Van, D., and Zant, H. Self-detecting gatetunable nanotube paddle resonators. Appl. Phys. Lett., 93(11), 111909(2008)
[26] Meyer, J. C., Paillet, M., and Roth, S. Single-molecule torsional pendulum. Science, 309(5740), 1539-1541(2005)
[27] Dong, L., Nelson, B. J., Fukuda, T., and Arai, F. Towards nanotube linear servomotors. IEEE Trans. Autom. Sci. Eng, 3(3), 228-235(2006)
[28] Williams, P., Papadakis, S., Patel, A., Falvo, M., Washburn, S., and Superfine, R. Torsional response and stiffening of individual multiwalled carbon nanotubes. Phys. Rev. Lett., 89(25), 255502(2002)
[29] Gheshlaghi, B. and Hasheminejad, S. M. Size dependent torsional vibration of nanotubes. Physica E, 43(1), 45-48(2010)
[30] Murmu, T., Adhikari, S., and Wang, C. Torsional vibration of carbon nanotube-buckyball systems based on nonlocal elasticity theory. Physica E, 43(6), 1276-1280(2011)
[31] Lim, C. W., Li, C., and Yu, J. Free torsional vibration of nanotubes based on nonlocal stress theory. J. Sound. Vib., 331(12), 2798-2808(2012)
[32] Loya, J., Aranda-Ruiz, J., and Fernández-Sáez, J. Torsion of cracked nanorods using a nonlocal elasticity model. J. Phys. D Appl. Phys., 47(11), 115304(2014)
[33] Arda, M. and Aydogdu, M. Torsional statics and dynamics of nanotubes embedded in an elastic medium. Compos. Struct., 114, 80-91(2014)
[34] Rao, S. S. Vibration of Continuous Systems, John Wiley and Sons, Hoboken (2007)
[35] Freund, L. and Herrmann, G. Dynamic fracture of a beam or plate in plane bending. J. Appl. Mech., 43(1), 112-116(1976)
[36] Miller, R. E. and Shenoy, V. B. Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 11(3), 139-147(2000)
[37] Assadi, A. and Farshi, B. Vibration characteristics of circular nanoplates. J. Appl. Phys., 108(7), 074312(2010)
[38] Hosseini-Hashemi, S. and Nazemnezhad, R. An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects. Compos. Part B-Eng., 52, 199-206(2013)
[39] Gheshlaghi, B. and Hasheminejad, S. M. Surface effects on nonlinear free vibration of nanobeams. Compos. Part B-Eng., 42, 934-937(2011)