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.
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.
4. Broek, D. The Practical Use of Fracture Mechanics. Kluwer Academic Publishers, Dordrecht, 1989.
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.
6. Challamel, N. On the comparison of Timoshenko and shear models in beam dynamics. J. Eng. Mech., 2006, 132(10), 1141−1145.
7. Challamel, N. Higher-order shear beam theories and enriched continuum. Mech. Res. Commun., 2011, 38(5), 388−392.
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.
9. Chondros, T. G., Dimarogonas, A. D. and Yao, J. A continuous cracked beam vibration theory. J. Sound Vib., 1998, 215(1), 17−34.
10. Dimarogonas, A. D. Vibration of cracked structures: A state of the art review. Eng. Fract. Mech., 1996, 55(5), 831−857.
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.
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.
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.
18. Pradhan, S. C. and Phadikar, J. K. Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib., 2009, 325, 206−223.
19. Reddy, J. N. Nonlocal theories of bending, buckling and vibration of beams. Int. J. Eng. Sci., 2007, 45(2−8), 288−307.
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.
21. Soedel, W. Vibrations of Shells and Plates. Marcel Dekker, New York, NY, 2004.
22. Tada, H., Paris, P. C. and Irwin, G. R. The Stress Analysis of Cracks Handbook. ASME Press, New York, NY, 2000.
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.
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.