1.1 Introduction
5
captured by a nonlocal parameter. The effects of nonlocal, point load and rampingtime parameters on the nanobeam vibrations were investigated.
Shafiei et al. [31] analysed transverse vibration of rotary functionally graded
size-dependent tapered Euler-Bernoulli nanobeam in thermal environment. Nonlocal
equations of motions were yielded by Hamilton’s principle, and they were solved
by the differential quadrature method. The following important parameters influence
on the nanobeams flapwise bending vibration were considered: angular velocity,
material distribution profile, boundary conditions, small-scale parameter and rate of
cross-sectional change.
Shahabinejad et al. [32] analysed free vibrations of rotating functionally graded
nanobeams under in-plane thermal loading. The Euler-Bernoulli beam theory, Hamilton’s principle and the small-scale effect based on the Eringen elasticity theory were
used. Free vibration frequencies were estimated for cantilever and proposed cantilever boundary conditions.
Arefi and Zenkour [33] employed the analytical approach for estimation of the
thermal stresses and deformations of a curved nanobeam resting on Pasternak’s
foundation. Influence of the following important parameters on the thermal stresses
was carried out: spring and shear parameters, thermal loads, nonlocal parameter, and
beam curvature radius.
1.1.1.3 Electric Field
McCutcheon et al. [34] demonstrated experimentally high-quality factor of dualpolarized photonic crystal nanobeam cavities.
Hu et al. [35] studied experimentally self-heating and external strain coupling
induced phase transition in a nanobeam. They demonstrated the accompanied gigantic change in resistivity and optical transmittance.
Liang and Shen [36] analysed the effect of an electrostatic force on an EulerBernoulli piezoelectric nanobeam dynamics. Influence of the electrostatic force on
the first four natural frequencies was demonstrated, and a possibility of adjusting the
natural frequency of a nanobeam by using voltage control was shown.
Stabile et al. [37] presented experimental results of the equilibrium/nonequilibrium transport properties of vanadium oxide nanobeams near the metalinsulator transition. It was shown a crucial role of both temperature and electric
fields in the transitional process, and that both fields can be separated.
Wang et al. [38] investigated vibration of nanowires based on the Timoshenko
beam model embedded in electric field via molecular dynamics simulation. It was
demonstrated an increase/decrease of the natural frequencies by increasing of positive/negative electric field in polarization direction. The vibration frequencies of the
cantilever Timoshenko beam with axial force were estimated versus the employed
electric field.
Phunpeng et al. [39] studied piezoelectric and flexoelectric effects of nanobeam
by using finite element method.
5
captured by a nonlocal parameter. The effects of nonlocal, point load and rampingtime parameters on the nanobeam vibrations were investigated.
Shafiei et al. [31] analysed transverse vibration of rotary functionally graded
size-dependent tapered Euler-Bernoulli nanobeam in thermal environment. Nonlocal
equations of motions were yielded by Hamilton’s principle, and they were solved
by the differential quadrature method. The following important parameters influence
on the nanobeams flapwise bending vibration were considered: angular velocity,
material distribution profile, boundary conditions, small-scale parameter and rate of
cross-sectional change.
Shahabinejad et al. [32] analysed free vibrations of rotating functionally graded
nanobeams under in-plane thermal loading. The Euler-Bernoulli beam theory, Hamilton’s principle and the small-scale effect based on the Eringen elasticity theory were
used. Free vibration frequencies were estimated for cantilever and proposed cantilever boundary conditions.
Arefi and Zenkour [33] employed the analytical approach for estimation of the
thermal stresses and deformations of a curved nanobeam resting on Pasternak’s
foundation. Influence of the following important parameters on the thermal stresses
was carried out: spring and shear parameters, thermal loads, nonlocal parameter, and
beam curvature radius.
1.1.1.3 Electric Field
McCutcheon et al. [34] demonstrated experimentally high-quality factor of dualpolarized photonic crystal nanobeam cavities.
Hu et al. [35] studied experimentally self-heating and external strain coupling
induced phase transition in a nanobeam. They demonstrated the accompanied gigantic change in resistivity and optical transmittance.
Liang and Shen [36] analysed the effect of an electrostatic force on an EulerBernoulli piezoelectric nanobeam dynamics. Influence of the electrostatic force on
the first four natural frequencies was demonstrated, and a possibility of adjusting the
natural frequency of a nanobeam by using voltage control was shown.
Stabile et al. [37] presented experimental results of the equilibrium/nonequilibrium transport properties of vanadium oxide nanobeams near the metalinsulator transition. It was shown a crucial role of both temperature and electric
fields in the transitional process, and that both fields can be separated.
Wang et al. [38] investigated vibration of nanowires based on the Timoshenko
beam model embedded in electric field via molecular dynamics simulation. It was
demonstrated an increase/decrease of the natural frequencies by increasing of positive/negative electric field in polarization direction. The vibration frequencies of the
cantilever Timoshenko beam with axial force were estimated versus the employed
electric field.
Phunpeng et al. [39] studied piezoelectric and flexoelectric effects of nanobeam
by using finite element method.