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Vibration of Initially Stressed Nonlocal Euler-Bernoulli Nano-Beams

Year 2018, Volume: 6 Issue: 1, 127 - 141, 31.01.2018
https://doi.org/10.21541/apjes.321997

Abstract

This paper is pertinent to the
analytical solutions for vibration analysis of initially stressed
Nonlocal Euler-Bernoulli nano-beams. In order to take into account of small length scale effect,
this vibration problem formulation is depending upon both nonlocal
Euler-Bernoulli and also Eringen’s nonlocal elasticity theory. The boundary
conditions and governing equation are obtained by use of Hamiltonian’s
principle. These equations are solved analytically with different initial
stresses (both compressive and tensile) and boundary conditions. The effect of small
length scale and the initial stress on the fundamental frequency are
investigated.  The solutions obtained are
compared with the ones depending upon both classical Euler-Bernoulli and
Timoshenko beam theory to comprehend the responses of nano-beams under the
effect of initial stress and small scale in terms of  frequencies for both theories. The results
supply a better declaration for vibration analysis of nano-beams which are
short and stubby with initial stress.



 

References

  • [1] S. Iijima, “Nanotubes”, Nature, vol. 354 pp. 56-58, 1999. [2] J. Mongillo, Nanotechnology 101, London: Greenwood, 2009. [3] M. Wilson, K. Kannangara, G. Smith, M. Simmons and B. Raguse, Nanotechnology, Basic Science and Emerging Technologies, Australia:Chapman&Hall/CRC, 2002. [4] L. F. Wang and H. Y. Hu, “Flexural wave propagation in single-walled carbon nanotubes”, Phys. Rev. B., vol. 7, pp. 195412, 2005. [5] A. C. Eringen, “Nonlocal polar elastic continua”, Int. J. Eng. Sci., vol. 10, pp. 1-16, 1972. [6] A. C. Eringen and D. Edelen, “On nonlocal elasticity”, Int. J. Eng. Sci., vol. 10, pp. 233-248, 1972. [7] A. C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves”, J. Appl. Phys., vol. 54, pp. 4703-4710, 1983. [8] J. Peddieson, G. R. Buchanan and R. P. McNitt, “Application of nonlocal continuum models to nanotechnology”, Int. J. Eng. Sci., vol. 41, pp. 305-312, 2003. [9] L. J. Sudak, “Column buckling of multiwalled carbon nanotubes using nonlocal continuum Mechanics”, J. Appl. Phys., vol. 94, pp. 7281-7287, 2003. [10] Y. Q. Zhang, G. R. Liu and X. Y. Xie, “Transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity”, Phys. Lett. A., vol. 340, pp. 258-266, 2005. [11] Q. Wang, “Wave propagation in carbon nanotubes via nonlocal continuum mechanics”, J. Appl. Phys., vol. 98, pp. 124301-124306, 2005. [12] Q. Wang and V. K. Varadan, Vibration of carbon nanotubes studied using nonlocal continuum mechanics, Smart. Mater. Struct., vol. 15, pp. 659-666, 2006. [13] P. Lu, H. P. Lee, C. Lu and P. Q. Zhang, “Dynamic properties of flexural beams using a nonlocal elasticity model”, J. Appl. Phys., vol. 99, pp. 073510, 2006. [14] M. Xu, “Free transverse vibrations of nano-to-micron scale beams”, Proceedings of the Royal Society A., vol. 462, pp. 2977-2995, 2006. [15] B. I. Yakobson, C. J. Brabec and J. Bernholc, “Nanomechanics of carbon nanotubes: Instability beyond linear response”, Phys. Rev. Lett., vol. 76, pp. 2511–2514, 1996. [16] C. M. Wang, Y. Y. Zhang and X. Q. He, “Vibration of nonlocal Timoshenko beams”, Nanotechnology, vol. 17, pp. 1–9, 2006. [17] Y. Y. Lee, C. M. Wang and S. Kitipornchai, “Vibration of Timoshenko beam with internal hinge”, J. Eng. Mech., vol. 129, pp. 293–301, 2003. [18] S. P. Timoshenko, Vibration Problems in Engineering, New York: Wiley, 1974. [19] P. Prasad, “On the response of a Timoshenko beam under initial stress to a moving load”, Int. J. Eng. Sci., vol. 19, pp. 615-628, 1981. [20] J. Yoon, C. Q. Ru and A. Mioduchowski, “Timoshenko-beam effects on transverse wave propagation in carbon nanotubes”, Compos. Part B-Eng., vol. 35, pp. 87-93, 2004. [21] S. J. A. Koh and H. P. Lee, “Molecular dynamics simulation of size and strain rate dependent mechanical response of FCC metallic nanowires”, Nanotechnology, vol. 17, pp. 3451–3467, 2006. [22] T. Murmu and S. Adhikari, “Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems”, European Journal of Mechanics A/Solids, vol. 34, pp. 52-62, 2012. [23] O. Rahmani, “On the flexural vibration of pre-stressed nanobeams based on a nonlocal theory”, Acta Physıca Polonıca A, vol. 125, pp. 532-533, 2014. [24] C. M. Wang, Y. Y. Zhang and S. Kitipornchai, “Vibration of initially stressed micro-and nano-beams”, Int. J. Struct. Stab. Dy., vol. 7, pp. 555-570, 2007. [25] B. N. Alemdar and P. Gülkan, “Beams on generalized foundations: supplementary element matrices”, Eng. Struct., vol. 19, pp. 910-920, 1997. [26] J. K. Phadikar and S. C. Pradhan, “Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates”, Comp. Mater. Sci., vol. 49, pp. 492-499, 2010. [27] S. C. Pradhan, “Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory”, Finite Elem. Anal. Des., vol. 50, pp. 8-20, 2012. [28] S. Adhikari, T. Murmu and M. A. McCarthy, “Dynamic finite element analysis of axially vibrating nonlocal rods”, Finite Elem. Anal. Des., vol. 63, pp. 42-50, 2013. [29] Ç. Dinçkal, B. N. Alemdar and P. H. Gülkan, “Dynamics of a beam-column element on an elastic foundation”, Can. J. Civ. Eng. vol. 43, pp. 685-701, 2016. [30] Ç. Dinçkal, Free vibration analysis of Carbon Nanotubes by using finite element method, Iran. J. Sci. Technol. Trans. Mech. Eng., vol. 40, pp. 43-55, 2016. [31] Ç. Dinçkal, “Finite element modeling for vibration of initially stressed nonlocal Euler- Bernoulli beams”, CBU J. of Sci., vol. 12, no 3, pp. 399-411, 2016.

Vibration of Initially Stressed Nonlocal Euler-Bernoulli Nano-Beams

Year 2018, Volume: 6 Issue: 1, 127 - 141, 31.01.2018
https://doi.org/10.21541/apjes.321997

Abstract

This paper is pertinent to the
analytical solutions for vibration analysis of initially stressed
Nonlocal Euler-Bernoulli nano-beams. In order to take into account of small length scale effect,
this vibration problem formulation is depending upon both nonlocal
Euler-Bernoulli and also Eringen’s nonlocal elasticity theory. The boundary
conditions and governing equation are obtained by use of Hamiltonian’s
principle. These equations are solved analytically with different initial
stresses (both compressive and tensile) and boundary conditions. The effect of small
length scale and the initial stress on the fundamental frequency are
investigated.  The solutions obtained are
compared with the ones depending upon both classical Euler-Bernoulli and
Timoshenko beam theory to comprehend the responses of nano-beams under the
effect of initial stress and small scale in terms of  frequencies for both theories. The results
supply a better declaration for vibration analysis of nano-beams which are
short and stubby with initial stress.

References

  • [1] S. Iijima, “Nanotubes”, Nature, vol. 354 pp. 56-58, 1999. [2] J. Mongillo, Nanotechnology 101, London: Greenwood, 2009. [3] M. Wilson, K. Kannangara, G. Smith, M. Simmons and B. Raguse, Nanotechnology, Basic Science and Emerging Technologies, Australia:Chapman&Hall/CRC, 2002. [4] L. F. Wang and H. Y. Hu, “Flexural wave propagation in single-walled carbon nanotubes”, Phys. Rev. B., vol. 7, pp. 195412, 2005. [5] A. C. Eringen, “Nonlocal polar elastic continua”, Int. J. Eng. Sci., vol. 10, pp. 1-16, 1972. [6] A. C. Eringen and D. Edelen, “On nonlocal elasticity”, Int. J. Eng. Sci., vol. 10, pp. 233-248, 1972. [7] A. C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves”, J. Appl. Phys., vol. 54, pp. 4703-4710, 1983. [8] J. Peddieson, G. R. Buchanan and R. P. McNitt, “Application of nonlocal continuum models to nanotechnology”, Int. J. Eng. Sci., vol. 41, pp. 305-312, 2003. [9] L. J. Sudak, “Column buckling of multiwalled carbon nanotubes using nonlocal continuum Mechanics”, J. Appl. Phys., vol. 94, pp. 7281-7287, 2003. [10] Y. Q. Zhang, G. R. Liu and X. Y. Xie, “Transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity”, Phys. Lett. A., vol. 340, pp. 258-266, 2005. [11] Q. Wang, “Wave propagation in carbon nanotubes via nonlocal continuum mechanics”, J. Appl. Phys., vol. 98, pp. 124301-124306, 2005. [12] Q. Wang and V. K. Varadan, Vibration of carbon nanotubes studied using nonlocal continuum mechanics, Smart. Mater. Struct., vol. 15, pp. 659-666, 2006. [13] P. Lu, H. P. Lee, C. Lu and P. Q. Zhang, “Dynamic properties of flexural beams using a nonlocal elasticity model”, J. Appl. Phys., vol. 99, pp. 073510, 2006. [14] M. Xu, “Free transverse vibrations of nano-to-micron scale beams”, Proceedings of the Royal Society A., vol. 462, pp. 2977-2995, 2006. [15] B. I. Yakobson, C. J. Brabec and J. Bernholc, “Nanomechanics of carbon nanotubes: Instability beyond linear response”, Phys. Rev. Lett., vol. 76, pp. 2511–2514, 1996. [16] C. M. Wang, Y. Y. Zhang and X. Q. He, “Vibration of nonlocal Timoshenko beams”, Nanotechnology, vol. 17, pp. 1–9, 2006. [17] Y. Y. Lee, C. M. Wang and S. Kitipornchai, “Vibration of Timoshenko beam with internal hinge”, J. Eng. Mech., vol. 129, pp. 293–301, 2003. [18] S. P. Timoshenko, Vibration Problems in Engineering, New York: Wiley, 1974. [19] P. Prasad, “On the response of a Timoshenko beam under initial stress to a moving load”, Int. J. Eng. Sci., vol. 19, pp. 615-628, 1981. [20] J. Yoon, C. Q. Ru and A. Mioduchowski, “Timoshenko-beam effects on transverse wave propagation in carbon nanotubes”, Compos. Part B-Eng., vol. 35, pp. 87-93, 2004. [21] S. J. A. Koh and H. P. Lee, “Molecular dynamics simulation of size and strain rate dependent mechanical response of FCC metallic nanowires”, Nanotechnology, vol. 17, pp. 3451–3467, 2006. [22] T. Murmu and S. Adhikari, “Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems”, European Journal of Mechanics A/Solids, vol. 34, pp. 52-62, 2012. [23] O. Rahmani, “On the flexural vibration of pre-stressed nanobeams based on a nonlocal theory”, Acta Physıca Polonıca A, vol. 125, pp. 532-533, 2014. [24] C. M. Wang, Y. Y. Zhang and S. Kitipornchai, “Vibration of initially stressed micro-and nano-beams”, Int. J. Struct. Stab. Dy., vol. 7, pp. 555-570, 2007. [25] B. N. Alemdar and P. Gülkan, “Beams on generalized foundations: supplementary element matrices”, Eng. Struct., vol. 19, pp. 910-920, 1997. [26] J. K. Phadikar and S. C. Pradhan, “Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates”, Comp. Mater. Sci., vol. 49, pp. 492-499, 2010. [27] S. C. Pradhan, “Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory”, Finite Elem. Anal. Des., vol. 50, pp. 8-20, 2012. [28] S. Adhikari, T. Murmu and M. A. McCarthy, “Dynamic finite element analysis of axially vibrating nonlocal rods”, Finite Elem. Anal. Des., vol. 63, pp. 42-50, 2013. [29] Ç. Dinçkal, B. N. Alemdar and P. H. Gülkan, “Dynamics of a beam-column element on an elastic foundation”, Can. J. Civ. Eng. vol. 43, pp. 685-701, 2016. [30] Ç. Dinçkal, Free vibration analysis of Carbon Nanotubes by using finite element method, Iran. J. Sci. Technol. Trans. Mech. Eng., vol. 40, pp. 43-55, 2016. [31] Ç. Dinçkal, “Finite element modeling for vibration of initially stressed nonlocal Euler- Bernoulli beams”, CBU J. of Sci., vol. 12, no 3, pp. 399-411, 2016.
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Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Çiğdem Dinçkal

Publication Date January 31, 2018
Submission Date June 16, 2017
Published in Issue Year 2018 Volume: 6 Issue: 1

Cite

IEEE Ç. Dinçkal, “Vibration of Initially Stressed Nonlocal Euler-Bernoulli Nano-Beams”, APJES, vol. 6, no. 1, pp. 127–141, 2018, doi: 10.21541/apjes.321997.