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Investigating various surface effects on nonlocal vibrational behavior of nanobeams

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Abstract

In this paper, the effects of various surface parameters on free vibration behavior of nanobeams are investigated in the presence of nonlocal effect. The Gurtin–Murdoch model is employed for incorporating the surface parameters including surface density, surface tension and surface elasticity, while the Eringen’s nonlocal elasticity theory takes into account the effect of small scales. The governing equations of motion are obtained for different materials such as aluminum and silicon with various boundary conditions. The high-precision semi-analytical differential transformation method is utilized to solve the governing equations. Then, by transforming the governing differential equations and boundary conditions into algebraic equations, the natural frequencies are obtained. The objective of the present study is to explore comprehensively the effect of different combination of various surface effects on nonlocal vibrational behavior of nanobeams for the first time. It is explicitly shown that the vibration characteristics of a nanobeam are significantly influenced by these surface effects. Moreover, it is shown that by increasing the size of beam, the influence of surface effects reduce to zero, and the natural frequency reaches its classical value. Numerical results are also presented to serve as benchmarks for future analyses of nanobeams.

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Abbreviations

\( \tau_{0}^{ \pm } \) :

Residual surface tensions under unconstrained conditions

\( \mu_{0}^{ \pm } ,\tau_{0}^{ \pm } \) :

Surface Lame constants on the surfaces S + and S −

\( \delta_{\alpha \beta } \) :

Kronecker delta

\( \rho_{0} \) :

Density of surface layer

E s :

Surface elasticity

τ 0 :

Surface tension

I s :

\( 2bh^{2} + \frac{{h^{3} }}{6} \) for a rectangular cross section

T :

Contact tractions on the contact surface between the bulk material and the surface layer

\( \ddot{u}_{i}^{s} \) :

Acceleration of the surface layer in the i-direction

C :

Fourth-order elasticity tensor

e o :

Material constant

a :

Internal characteristic length

μ :

Nonlocal parameter

t :

Macroscopic stress at a point

ε :

Strain at the point

σ :

Stress at the point

E :

Young’s modulus

h :

Height of the beam

L :

Length of beam

b :

Width of beam

(u, w):

Axial and transverse displacements of a point

K :

Kinetic energy

U :

Strain energy

V :

Work done by external forces

ρ :

Mass density

I :

Moment inertia

A :

Cross-sectional area

f :

External axial loads along length of beam

q :

External transverse loads along length of beam

ν :

Poisson’s ratio

γ :

Transverse shear strain

M :

Moment

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran

    Farzad Ebrahimi & Mahya Boreiry

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  1. Farzad Ebrahimi
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  2. Mahya Boreiry
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Correspondence to Farzad Ebrahimi.

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Ebrahimi, F., Boreiry, M. Investigating various surface effects on nonlocal vibrational behavior of nanobeams. Appl. Phys. A 121, 1305–1316 (2015). https://doi.org/10.1007/s00339-015-9512-6

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