Surface Energy Effect on Free Axial Vibration of Cracked Nanorods Made of Functionally Graded Materials Based on Rayleigh Theory

Authors

1 School of Engineering, Damghan University, Damghan, Iran

2 Department of Engineering, Kharazmi University, Tehran, Iran

Abstract

The aim of this research is investigation of surface enerfy effet on free axial vibration of cracked functionally graded nanorods modelled based on the Rayleigh theory of rods. In Rayleigh theory, the effect not only of the axial inertia but also of the inertia of lateral motions are considered. It is assumed that the material of nanorod is functionally graded in its length direction and varies as power low relation. The crack is also modelled as a linear spring in which its stiffness is proporsional to crack severity. The surface energy is included effects of the surface density, surface stress and surface Lame constants parameters. Due to considering the effect of surface energy parameters, the governing equations of motions and corresponding boundary conditions become inhomogeneous in which to solve them, they are converted to homogeneus ones using an appropriate change of variable, firstly. Then, the natural frequencies of fixed-fixed and fixed-free nanorods are extracted using the method of harmonic differential quadrature. In addition to type of boundary condition, effects of parameters like length and radius of nanorod, severe and location of crack, and mode number on natural axial frequencies of cracked functionally graded nanorods in presence of surface energy are investigated.

Keywords

Main Subjects

References

[1] Assadi A, Farshi B (2011) Size-dependent longitudinal and transverse wave propagation in embedded nanotubes with consideration of surface effects. Acta Mech 222(1): 27-39.
[2] Hosseini-Hashemi S, Fakher M, Nazemnezhad R (2017) Longitudinal vibrations of aluminum nanobeams by applying elastic moduli of bulk and surface: molecular dynamics simulation and continuum model. Mater Res Express 4(8): 085036.
[3] Nazemnezhad R, Mahoori R, Samadzadeh A (2019) Surface energy effect on nonlinear free axial vibration and internal resonances of nanoscale rods. Eur J Mech A Solids 77: 103784.
[4] Nazemnezhad R, Shokrollahi H (2019) Free axial vibration analysis of functionally graded nanorods using surface elasticity theory. Modares Mech Eng 18(9): 131-141.
[5] Nazemnezhad R, Shokrollahi H (2020) Free axial vibration of cracked axially functionally graded nanoscale rods incorporating surface effect. Steel Compos Struct 35(3): 449-462.
[6] Karliؤچiؤ‡ DZ, Ayed S, Flaieh E (2019) Nonlocal axial vibration of the multiple Bishop nanorod system. Math Mech Solids 24(6): 1668-1691.
[7] Babaei A (2019) Longitudinal vibration responses of axially functionally graded optimized MEMS gyroscope using Rayleigh–Ritz method, determination of discernible patterns and chaotic regimes. SN App Sci 1(8): 831.
[8] Yayli Mأ– (2018) Free longitudinal vibration of a nanorod with elastic spring boundary conditions made of functionally graded material. Micro Nano Lett 13(7): 1031-1035.
[9] Nazemnezhad R, Kamali K (2018) Free axial vibration analysis of axially functionally graded thick nanorods using nonlocal Bishop's theory. Steel Compos Struct 28(6): 749-758.
[10] Aydogdu M, Arda M, Filiz S (2018) Vibration of axially functionally graded nano rods and beams with a variable nonlocal parameter. Adv nano res 6(3): 257.
[11] Akgأ¶z B, Civalek أ– (2013) Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM). Compos Part B-Eng 55: 263-268.
[12] إ‍imإںek M (2012) Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods. Comp Mater Sci 61: 257-265.
[13] Rao SS. Vibration of continuous systems: Wiley Online Library, 2007.
[14] Yayli Mأ– (2020) Axial vibration analysis of a Rayleigh nanorod with deformable boundaries. Microsyst Technol: 26:2661-2671.
[15] Civalek أ– (2004) Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Eng Struct 26(2): 171-186.
Journal of Solid and Fluid Mechanics
Volume 12, Issue 4
September and October 2022
Pages 103-116

Files

Share

How to cite

Statistics