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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.
beam, crack, non-local elasticity, nano-material, vibration
[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. https://doi.org/10.1063/1.332803
[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. https://doi.org/10.1016/j.ijengsci.2007.04.004
[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.https://doi.org/10.1007/s10483-010-0105-7
[6] Emam, S.A., A general nonlocal nonlinear model for buckling of nanobeams. Applied Mathematics Modelling, 37(10–11), pp. 6929–6939, 2013. https://doi.org/10.1016/j.apm.2013.01.043
[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. https://doi.org/10.1016/j.mechrescom.2010.05.007
[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. https://doi.org/10.1142/s0219455411004087
[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. https://doi.org/10.1063/1.2189213
[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. https://doi.org/10.1016/j.compstruct.2012.08.024
[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. https://doi.org/10.1016/j.apm.2013.08.011
[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.https://doi.org/10.1016/0013-7944(94)00175-8
[14] Lellep, J. & Kraav, T., Buckling of beams and columns with defects. International Journal of Structural Stability and Dynamics, 16(8), 2016. https://doi.org/10.1142/s0219455415500480
[15] Lellep, J. & Liyvapuu, A., Natural vibrations of stepped arches with cracks. Agronomy Research, 14(3), pp. 821–830, 2016.