A Solution Method for Longitudinal Vibrations of Functionally Graded Nanorods

Abstract

In the present study, a nonlocal finite element formulation of free longitudinal vibration is derived for functionally graded nano-sized rods. Size dependency is considered via Eringen’s nonlocal elasticity theory. Material properties, Young’s modulus and mass density, of the nano-sized rod change in the thickness direction according to the power-law. For the examined FG nanorod finite element, the axial displacement is specified with a linear function. The stiffness and mass matrices of functionally graded nano-sized rod are found by means of interpolation functions. Functionally graded nanorod is considered with clamped-free boundary condition and its longitudinal vibration analysis is performed.

Keywords

• Nonlocal elaticity theory
• Functionally graded materials
• Nanorod
• Finite element method
• Vibration

References

1. [1] Lü, C.F., Lim, C.W., Chen, W.Q., Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. International Journal of Solids and Structures, 46(5), 1176-1185, 2009.
2. [2] Lanhe, W., Thermal buckling of a simply supported moderately thick rectangular FGM plate. Composite Structures, 64(2), 211-218, 2004.
3. [3] Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E.A., Mahmoud, S.R., On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model. Steel Compos. Struct, 18(4), 1063-1081, 2015.
4. [4] Żur, K.K., Arefi, M., Kim, J., Reddy, J.N. Free vibration and buckling analyses of magneto-electro-elastic FGM nanoplates based on nonlocal modified higher-order sinusoidal shear deformation theory. Composites Part B: Engineering, 182, 107601, 2020.
5. [5] Ebrahimi, F., Ehyaei, J., Babaei, R. Thermal buckling of FGM nanoplates subjected to linear and nonlinear varying loads on Pasternak foundation. Advances in materials Research, 5(4), 245, 2016.
6. [6] Yuan, Y., Zhao, K., Sahmani, S., Safaei, B. Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes. Applied Mathematics and Mechanics, 1-18, 2020.
7. [7] Karami, B., Shahsavari, D., Janghorban, M., Li, L. On the resonance of functionally graded nanoplates using bi-Helmholtz nonlocal strain gradient theory. International Journal of Engineering Science, 144, 103143, 2019.
8. [8] Uzun, B., Yaylı, M.Ö. Nonlocal vibration analysis of Ti-6Al-4V/ZrO2 functionally graded nanobeam on elastic matrix. Arabian Journal of Geosciences, 13(4), 1-10, 2020.

9. [9] Uzun, B., Yaylı, M. Ö., Deliktaş, B. Free vibration of FG nanobeam using a finite-element method. Micro & Nano Letters, 15(1), 35-40, 2020.
10. [10] Uzun, B., Yaylı, M. Ö., Finite element model of functionally graded nanobeam for free vibration analysis. International Journal of Engineering and Applied Sciences, 11(2), 387-400, 2019.
11. [11] Hosseini, S.A.H., Rahmani, O., Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model. Applied Physics A, 122(3), 169, 2016.
12. [12] Jalaei, M.H., Civalek, Ӧ., On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam. International Journal of Engineering Science, 143, 14-32, 2019.
13. [13] Saffari, S., Hashemian, M., Toghraie, D., Dynamic stability of functionally graded nanobeam based on nonlocal Timoshenko theory considering surface effects. Physica B: Condensed Matter, 520, 97-105, 2017.
14. [14] Aydogdu, M., Arda, M., Filiz, S., Vibration of axially functionally graded nano rods and beams with a variable nonlocal parameter. Advances in nano research, 6(3), 257, 2018.
15. [15] Arda, M., Axial dynamics of functionally graded Rayleigh-Bishop nanorods. Microsystem Technologies, 1-14, 2020.
16. [16] Kiani, K., Free dynamic analysis of functionally graded tapered nanorods via a newly developed nonlocal surface energy-based integro-differential model. Composite Structures, 139, 151-166, 2016.
17. [17] Arefi, M., Zenkour, A. M., Employing the coupled stress components and surface elasticity for nonlocal solution of wave propagation of a functionally graded piezoelectric Love nanorod model. Journal of Intelligent Material Systems and Structures, 28(17), 2403-2413, 2017.
18. [18] Narendar, S., Wave dispersion in functionally graded magneto-electro-elastic nonlocal rod. Aerospace Science and Technology, 51, 42-51 2016.
19. [19] Demir, Ç., Civalek, Ö., A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Composite Structures, 168, 872-884, 2017.
20. [20] Adhikari, S., Murmu, T., McCarthy, M.A., Dynamic finite element analysis of axially vibrating nonlocal rods. Finite Elements in Analysis and Design, 63, 42-50, 2013.
21. [21] Hemmatnezhad, M., Ansari, R., Finite element formulation for the free vibration analysis of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory. Journal of theoretical and applied physics, 7(1), 6, 2013.
22. [22] Civalek, Ö., Uzun, B., Yaylı, M.Ö., Akgöz, B. Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method. The European Physical Journal Plus, 135(4), 381, 2020.
23. [23] Ghannadpour, S. A. M. (2019). A variational formulation to find finite element bending, buckling and vibration equations of nonlocal Timoshenko beams. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 43(1), 493-502.
24. [24] Akbaş, Ş.D., Static, Vibration, and Buckling Analysis of Nanobeams. Nanomechanics, 123, 2017.
25. [25] Anjomshoa, A., Shahidi, A. R., Hassani, B., Jomehzadeh, E. Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory. Applied Mathematical Modelling, 38(24), 5934-5955, 2014.
26. [26] Taghizadeh, M., Ovesy, H.R., Ghannadpour, S.A.M. Beam buckling analysis by nonlocal integral elasticity finite element method. International Journal of Structural Stability and Dynamics, 16(06), 1550015, 2016.
27. [27] Demir, C., Mercan, K., Numanoglu, H.M., Civalek, O., Bending response of nanobeams resting on elastic foundation. Journal of Applied and Computational Mechanics, 4(2), 105-114, 2018.
28. [28] Taghizadeh, M., Ovesy, H.R., Ghannadpour, S.A.M., Nonlocal integral elasticity analysis of beam bending by using finite element method. Structural Engineering and Mechanics, 54(4), 755-769, 2015.
29. [29] Mahmoud, F.F., Eltaher, M.A., Alshorbagy, A.E., Meletis, E.I., Static analysis of nanobeams including surface effects by nonlocal finite element. Journal of mechanical science and technology, 26(11), 3555-3563, 2012.
30. [30] Reddy, J.N., Energy Principles and Variational Methods in Applied Mechanics (2nd ed.)’ (John Wiley & Sons, New York, 2002).
31. [31] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of applied physics, 54(9), 4703-4710, 1983.
32. [32] Xu, X.J., Zheng, M.L., Wang, X.C. On vibrations of nonlocal rods: Boundary conditions, exact solutions and their asymptotics. International Journal of Engineering Science, 119, 217-231, 2017.
33. [33] Numanoğlu, H.M., Akgöz, B., Civalek, Ö. On dynamic analysis of nanorods. International Journal of Engineering Science, 130, 33-50, 2018.
34. [34] Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T., The modified couple stress functionally graded Timoshenko beam formulation. Materials & Design, 32(3), 1435-1443, 2011.