ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
Proceeding cover
proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Free vibrations of stepped nano-beams with cracks; pp. 103–116
PDF | 10.3176/proc.2022.1.09

Authors
Jaan Lellep, Artur Lenbaum
Abstract

Free vibrations of stepped beams made of nano-materials are investigated. The nano-beams under consideration have piecewise constant dimensions of the cross section and are weakened with crack-like defects. The influence of the crack on the mechanical behaviour of the nano-beam is defined with the aid of the additional local compliance. Mechanical vibrations of the nano-beam are specified in the non-local theory of elasticity developed by Eringen. Numerical results are presented for nano-beams with two different thicknesses.

References

1. Adali, S. Variational formulation for buckling of multi-walled carbon nanotubes modelled as nonlocal Timoshenko beams. J. Theor. Appl. Mech., 2012, 50(1), 321−333.

2. Anderson, T. L. Fracture Mechanics. Fundamentals and Applications. CRC Press, Boca Raton, FL, 2005.

3. Anifantis, N. and Dimarogonas, A. Stability of columns with a single crack subjected to follower and vertical loads. Int. J. Solids Struct., 1983, 19(4), 281−291.
https://doi.org/10.1016/0020-7683(83)90027-6

4. Broek, D. The Practical Use of Fracture Mechanics. Kluwer Academic Publishers, Dordrecht, 1989.
https://doi.org/10.1007/978-94-009-2558-8

5. Caddemi, S. and Caliò, I. The influence of the axial force on the vibration of the Euler−Bernoulli beam with an arbitrary number of cracks. Arch. Appl. Mech., 2012, 82(6), 827−839.
https://doi.org/10.1007/s00419-011-0595-z

6. Challamel, N. On the comparison of Timoshenko and shear models in beam dynamics. J. Eng. Mech., 2006, 132(10), 1141−1145.
https://doi.org/10.1061/(ASCE)0733-9399(2006)132:10(1141)

7. Challamel, N. Higher-order shear beam theories and enriched continuum. Mech. Res. Commun., 2011, 38(5), 388−392.
https://doi.org/10.1016/j.mechrescom.2011.05.004

8. Challamel, N. and Elishakoff, I. Surface stress effects may induce softening: Euler−Bernoulli and Timoshenko buckling solutions. Physica E, 2012, 44(9), 1862−1867.
https://doi.org/10.1016/j.physe.2012.05.019

9. Chondros, T. G., Dimarogonas, A. D. and Yao, J. A continuous cracked beam vibration theory. J. Sound Vib., 1998, 215(1), 17−34.
https://doi.org/10.1006/jsvi.1998.1640

10. Dimarogonas, A. D. Vibration of cracked structures: A state of the art review. Eng. Fract. Mech., 1996, 55(5), 831−857.
https://doi.org/10.1016/0013-7944(94)00175-8

11. Eringen, A. C. Nonlocal Continuum Field Theories. Springer, New York, NY, 2002.

12. Ghannadpour, S. A. M., Mohammadi, B. and Fazilati, J. Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method. Compos. Struct., 2013, 96, 584−589.
https://doi.org/10.1016/j.compstruct.2012.08.024

13. Hellan, K. Introduction to Fracture Mechanics. McGraw-Hill, New York, NY, 1984.

14. Lellep, J. and Kraav, T. Buckling of beams and columns with defects. Int. J. Struct. Stab. Dyn., 2016, 16(8), 2550048.
https://doi.org/10.1142/S0219455415500480

15. Lellep, J. and Kägo, E. Vibrations of elastic stretched strips with cracks. Int. J. Mech., 2011, 5(1), 27−34.

16. Lellep, J. and Lenbaum, A. Natural vibrations of a nano-beam with cracks. Int. J. Theor. Appl. Mech., 2016, 1(1), 247−252.

17. Lu, P., Lee, H. P. and Lu, C. Dynamic properties of flexural beams using a nonlocal elasticity model. J. Appl. Phys., 2006, 99, 073510.
https://doi.org/10.1063/1.2189213

18. Pradhan, S. C. and Phadikar, J. K. Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib., 2009, 325, 206−223.
https://doi.org/10.1016/j.jsv.2009.03.007

19. Reddy, J. N. Nonlocal theories of bending, buckling and vibration of beams. Int. J. Eng. Sci., 2007, 45(2−8), 288−307.
https://doi.org/10.1016/j.ijengsci.2007.04.004

20. Roostai, H. and Haghpanahi, M. Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory. Appl. Math. Model., 2014, 38(3), 1159−1169. 
https://doi.org/10.1016/j.apm.2013.08.011

21. Soedel, W. Vibrations of Shells and Plates. Marcel Dekker, New York, NY, 2004.
https://doi.org/10.4324/9780203026304

22. Tada, H., Paris, P. C. and Irwin, G. R. The Stress Analysis of Cracks Handbook. ASME Press, New York, NY, 2000.
https://doi.org/10.1115/1.801535

23. Wang, C. M., Zhang, Y. Y., Ramesh, S. S. and Kitipornchai, S. Buckling analysis of micro- and nanorods/tubes based on nonlocal Timoshenko beam theory. J. Phys. D: Appl. Phys., 2006, 39, 3904−3909.
https://doi.org/10.1088/0022-3727/39/17/029

24. Zhou, L. and Huang, Y. Crack effect on the elastic buckling behavior of axially and eccentrically loaded columns. Struct. Eng. Mech., 2006, 22(2), 169−184. 
https://doi.org/10.12989/sem.2006.22.2.169
 

Back to Issue