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[1] Eringen, A.C., Nonlocal Continuum Field Theories. Springer, New York, 2002.
[2] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal Of Applied Physics, 54(9), pp. 4703–4710, 1983. [Crossref]
[3] Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45(2–8), pp. 288–307, 2007. [Crossref]
[4] Lim, C.W., Equilibrium and static deflection for bending of a nonlocal nanobeam. Advances in Vibration Engineering, 8, pp. 277–300, 2009.
[5] Lim, C.W., On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Applied Mathematics and Mechanics, 31(1), pp. 37–54, 2010. [Crossref]
[6] Emam, S.A., A general nonlocal nonlinear model for buckling of nanobeams. Applied Mathematics Modelling, 37(10–11), pp. 6929–6939, 2013. [Crossref]
[7] Challamel, N., Meftah, S.A. & Bernard,F., Buckling of elastic beams on non-local foundation: A revisiting of Reissner model. Mechanics Research Communications, 37(5), pp. 472–475, 2010. [Crossref]
[8] Li, C., Lim, C.W., Yu, J.L. & Zheng, O.C., Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. International Journal Structural Stability and Dynamics, 11(2), pp. 257–271, 2011. [Crossref]
[9] Lu, P., Lee, H.P. & Lu, C., Dynamic properties of flexural beams using a nonlocal elasticity model. Journal Applied Physics, 99(7), p. 073510, 2006. [Crossref]
[10] Ghannadpour, S.A.M. & Fazilati, B.M., Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method. Composite Structures, 96, pp. 584–589, 2013. [Crossref]
[11] Roostai, H. & Haghpanahi, M., Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory. Applied Mathematical Modelling, 38(3), pp. 1159–1169, 2014. [Crossref]
[12] Lellep, J. & Lenbaum, A., Natural vibrations of a nano-beam with cracks. International Journal of Theoretical and Applied Mechanics, 1(1), pp. 247–252, 2016.
[13] Dimarogonas, A.D., Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics, 55(5), pp. 831–857, 1996. [Crossref]
[14] Lellep, J. & Kraav, T., Buckling of beams and columns with defects. International Journal of Structural Stability and Dynamics, 16(8), 2016. [Crossref]
[15] Lellep, J. & Liyvapuu, A., Natural vibrations of stepped arches with cracks. Agronomy Research, 14(3), pp. 821–830, 2016.
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Open Access
Research article

Free Vibrations of Stepped Nano-Beams

Jaan Lellep,
Artur Lenbaum
Institute of Mathematics and Statistics, University of Tartu, Estonia
International Journal of Computational Methods and Experimental Measurements
|
Volume 6, Issue 4, 2018
|
Pages 716-725
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: N/A
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Abstract:

Free vibrations of beams and rods made of nano-materials are investigated. It is assumed that the dimensions of cross sections of nano-beams are piecewise constant and that the beams are weakened with cracks. It is expected that the vibrational behaviour of the nano-material can be described within the non-local theory of elasticity and that the crack induces additional local compliance. The latter is coupled with the stress intensity coefficient at the crack tip.

Keywords: Beam, Crack, Non-local elasticity, Nano-material, Vibration
Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References
[1] Eringen, A.C., Nonlocal Continuum Field Theories. Springer, New York, 2002.
[2] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal Of Applied Physics, 54(9), pp. 4703–4710, 1983. [Crossref]
[3] Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45(2–8), pp. 288–307, 2007. [Crossref]
[4] Lim, C.W., Equilibrium and static deflection for bending of a nonlocal nanobeam. Advances in Vibration Engineering, 8, pp. 277–300, 2009.
[5] Lim, C.W., On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Applied Mathematics and Mechanics, 31(1), pp. 37–54, 2010. [Crossref]
[6] Emam, S.A., A general nonlocal nonlinear model for buckling of nanobeams. Applied Mathematics Modelling, 37(10–11), pp. 6929–6939, 2013. [Crossref]
[7] Challamel, N., Meftah, S.A. & Bernard,F., Buckling of elastic beams on non-local foundation: A revisiting of Reissner model. Mechanics Research Communications, 37(5), pp. 472–475, 2010. [Crossref]
[8] Li, C., Lim, C.W., Yu, J.L. & Zheng, O.C., Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. International Journal Structural Stability and Dynamics, 11(2), pp. 257–271, 2011. [Crossref]
[9] Lu, P., Lee, H.P. & Lu, C., Dynamic properties of flexural beams using a nonlocal elasticity model. Journal Applied Physics, 99(7), p. 073510, 2006. [Crossref]
[10] Ghannadpour, S.A.M. & Fazilati, B.M., Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method. Composite Structures, 96, pp. 584–589, 2013. [Crossref]
[11] Roostai, H. & Haghpanahi, M., Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory. Applied Mathematical Modelling, 38(3), pp. 1159–1169, 2014. [Crossref]
[12] Lellep, J. & Lenbaum, A., Natural vibrations of a nano-beam with cracks. International Journal of Theoretical and Applied Mechanics, 1(1), pp. 247–252, 2016.
[13] Dimarogonas, A.D., Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics, 55(5), pp. 831–857, 1996. [Crossref]
[14] Lellep, J. & Kraav, T., Buckling of beams and columns with defects. International Journal of Structural Stability and Dynamics, 16(8), 2016. [Crossref]
[15] Lellep, J. & Liyvapuu, A., Natural vibrations of stepped arches with cracks. Agronomy Research, 14(3), pp. 821–830, 2016.
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Lellep, J. & Lenbaum, A. (2018). Free Vibrations of Stepped Nano-Beams. Int. J. Comput. Methods Exp. Meas., 6(4), 716-725. https://doi.org/10.2495/CMEM-V6-N4-716-725
J. Lellep and A. Lenbaum, "Free Vibrations of Stepped Nano-Beams," Int. J. Comput. Methods Exp. Meas., vol. 6, no. 4, pp. 716-725, 2018. https://doi.org/10.2495/CMEM-V6-N4-716-725
@research-article{Lellep2018FreeVO,
title={Free Vibrations of Stepped Nano-Beams},
author={Jaan Lellep and Artur Lenbaum},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2018},
page={716-725},
doi={https://doi.org/10.2495/CMEM-V6-N4-716-725}
}
Jaan Lellep, et al. "Free Vibrations of Stepped Nano-Beams." International Journal of Computational Methods and Experimental Measurements, v 6, pp 716-725. doi: https://doi.org/10.2495/CMEM-V6-N4-716-725
Jaan Lellep and Artur Lenbaum. "Free Vibrations of Stepped Nano-Beams." International Journal of Computational Methods and Experimental Measurements, 6, (2018): 716-725. doi: https://doi.org/10.2495/CMEM-V6-N4-716-725
LELLEP J， LENBAUM A. Free Vibrations of Stepped Nano-Beams[J]. International Journal of Computational Methods and Experimental Measurements， 2018， 6(4)： 716-725. https://doi.org/10.2495/CMEM-V6-N4-716-725