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Research Papers

Transverse Vibrations of Mixed-Mode Cracked Nanobeams With Surface Effect

[+] Author and Article Information
Kai-Ming Hu

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
800 Dongchuan Road,
Shanghai 200240, China
e-mail: hukaiming@sjtu.edu.cn

Wen-Ming Zhang

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
800 Dongchuan Road,
Shanghai 200240, China
e-mail: wenmingz@sjtu.edu.cn

Zhi-Ke Peng

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
800 Dongchuan Road,
Shanghai 200240, China
e-mail: z.peng@sjtu.edu.cn

Guang Meng

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
800 Dongchuan Road,
Shanghai 200240, China
e-mail: gmeng@sjtu.edu.cn

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 12, 2014; final manuscript received October 12, 2015; published online November 24, 2015. Assoc. Editor: Nader Jalili.

J. Vib. Acoust 138(1), 011020 (Nov 24, 2015) (12 pages) Paper No: VIB-14-1471; doi: 10.1115/1.4031832 History: Received December 12, 2014; Revised October 12, 2015; Accepted October 13, 2015

Slant edge cracked effect considering the inherent relation between surface energy and mixed-mode crack propagations on the free transverse vibrations of nanobeams with surface effect is investigated. First, the slant edge cracked effect, which considers residual surface stress effect on the crack tip fields of a mode-I and mode-II surface edge crack, is developed and the corresponding stress intensity factors (SIFs) and local flexibility coefficients are derived. Moreover, a refined continuum model of slant cracked nanobeams is established by considering both slant edge cracked effect and surface effect. The effects of fracture angles, crack depth, surface elasticity, surface stress, and surface density on the local flexibility and free transverse vibration characteristics of cracked nanobeams are, respectively, analyzed. The results show that the flexibility coefficients distribute symmetrically about residual surface stress. Fracture angles have a profound influence on both the symmetries of the mode shapes and the natural frequencies of nanobeams, and the influence becomes more pronounced as crack depth ratios increase. Furthermore, the natural frequencies will first decrease and then increase with fracture angles when the slant edge cracked effect is considered. The results demonstrate that the inherent relation between surface energy and crack propagations should be considered for both the stress distributions at the crack tip and the dynamic behavior of cracked nanobeams.

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Figures

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Fig. 1

(a) Free-body of a nanobeam element with a rectangular cross section and a slant edge crack oriented at an angle θ on the surface of the beam and stress components σs,τs resolved by normal force σ0 due to the bending moment M; (b) slant cracks due to mechanical fatigue reported by Pasquale and Soma (© 2015 IEEE. Reprinted, with permission, from Ref. [35]).

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Fig. 2

Schematic view of an infinite medium with a slant surface inside crack of length 2α and fracture angles θ

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Fig. 3

Variations of the additional flexibility coefficients of the slant cracked nanobeams with respect to: (a) residual surface stress τ0 under different fracture angles and (b) depth ratios γ under different residual surface stresses

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Fig. 4

First four mode shapes of clamped–clamped nanobeams with slant edge cracked effect and surface effect for different depth ratios and θ=45 deg, X1=0.5

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Fig. 5

Values of the first nondimensional natural frequency of clamped–clamped nanobeams with mixed-mode cracks and surface effect as a function of fracture angles for different depth ratios

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Fig. 6

The first nondimensional natural frequency ratio of clamped–clamped cracked nanobeams with respect to: (a) depth ratios and (b) depth ratios and fracture angles for different cases of surface effect, where KI0s,KII0s denote the SIFs of a mixed-mode crack without residual surface stress effect

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Fig. 7

The first natural frequencies of clamped–clamped cracked nanobeams as a function of depth ratios and fracture angles for different cases of residual surface stress: (a) τ0=0 N/m, (b) τ0 = 10 N/m, (c) τ0 = 20 N/m, and (d) τ0 = 50 N/m

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Fig. 8

Variations of the first nondimensional natural frequencies of clamped–clamped nanobeams with mixed-mode cracks under different surface effect defined in Table 2 and L = 100 nm: (a) with γ=0.2 as a function of fracture angles and (b) with θ = 45 deg as a function of depth ratios

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Fig. 9

Surface effect on the first nondimensional natural frequencies of clamped–clamped cracked nanobeams: (a) the frequency ratio with respect to surface density and (b) the frequency ratio with respect to surface stress for different surface elasticity

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