1.1 Introduction
3
boundary conditions. They also illustrated how the nanoscale effects change the
Timoshenko beam natural frequencies.
Sahmani and Aghdam [13] investigated the size-dependent nonlinear free vibration response of multi-layer functionally graded graphene platelet-reinforced composite nanobeams. Both of the hardening/softening stiffness were taken into considerations within the framework of the third-order shear deformation beam theory
and nonlocal strain gradient elasticity theory. The non-classical governing PDEs
were obtained, and next a perturbation technique in conjunction with the Galerkin
method yielded an explicit analytical solution for nonlocal strain gradient nonlinear
frequency of the studied nanobeams.
Jena and Chakraverty [14] studied the free vibration of nanobeams based on
nonlocal Euler-Bernoulli theory and the developed differential transform method.
The latter allowed to transform the governing differential equations to the algebraic
equations. The numerical results for different scaling parameters and four boundary
conditions were presented and discussed.
Khaniki [15] studied vibrations of nanobeams based on the modified Eringen’s
two-phase local/nonlocal integral model. Three different examples including inphase vibration, out-phase vibration and fixation of the under heath beam layer
were analysed. It was shown, among others, that the elastic coupling term and nonlocal parameters had a significant effect on the natural frequencies of the studied
nanobeams.
1.1.1.2 Thermal Environment
Jiang et al. [16] determined the thermal extension coefficient of carbon nanotubes
using an analytical approach.
Yan and Han [17] investigated the torsional and axially compressed buckling of
multi-walled and double-walled carbon nanotubes versus temperature change.
Lee and Chang [18] found a closed-form solution while investigating the critical
buckling temperature of single-walled carbon nanotubes under a uniform temperature
rise.
Tounsi et al. [19] studied the small size effects on wave propagation in doublewalled carbon nanotube subjected to temperature.
Lim and Yang [20] employed the variational principle and integrated the straining energy density of a nanobeam under thermal field based on originally developed higher order differential equation and the corresponding boundary conditions.
The effects of the nonlocal nanoscale and temperature on the nanobeams transverse
deflection were illustrated and analysed. It was concluded that at low and room temperature the nanobeams’ transverse deflection decreased with increase in temperature
difference, whereas at higher temperature the transverse deflections increased as the
temperature difference increased.
Chen et al. [21] predicted the damping behaviour of the nanobeams including
thermal fluctuations and the paddling effect.
3
boundary conditions. They also illustrated how the nanoscale effects change the
Timoshenko beam natural frequencies.
Sahmani and Aghdam [13] investigated the size-dependent nonlinear free vibration response of multi-layer functionally graded graphene platelet-reinforced composite nanobeams. Both of the hardening/softening stiffness were taken into considerations within the framework of the third-order shear deformation beam theory
and nonlocal strain gradient elasticity theory. The non-classical governing PDEs
were obtained, and next a perturbation technique in conjunction with the Galerkin
method yielded an explicit analytical solution for nonlocal strain gradient nonlinear
frequency of the studied nanobeams.
Jena and Chakraverty [14] studied the free vibration of nanobeams based on
nonlocal Euler-Bernoulli theory and the developed differential transform method.
The latter allowed to transform the governing differential equations to the algebraic
equations. The numerical results for different scaling parameters and four boundary
conditions were presented and discussed.
Khaniki [15] studied vibrations of nanobeams based on the modified Eringen’s
two-phase local/nonlocal integral model. Three different examples including inphase vibration, out-phase vibration and fixation of the under heath beam layer
were analysed. It was shown, among others, that the elastic coupling term and nonlocal parameters had a significant effect on the natural frequencies of the studied
nanobeams.
1.1.1.2 Thermal Environment
Jiang et al. [16] determined the thermal extension coefficient of carbon nanotubes
using an analytical approach.
Yan and Han [17] investigated the torsional and axially compressed buckling of
multi-walled and double-walled carbon nanotubes versus temperature change.
Lee and Chang [18] found a closed-form solution while investigating the critical
buckling temperature of single-walled carbon nanotubes under a uniform temperature
rise.
Tounsi et al. [19] studied the small size effects on wave propagation in doublewalled carbon nanotube subjected to temperature.
Lim and Yang [20] employed the variational principle and integrated the straining energy density of a nanobeam under thermal field based on originally developed higher order differential equation and the corresponding boundary conditions.
The effects of the nonlocal nanoscale and temperature on the nanobeams transverse
deflection were illustrated and analysed. It was concluded that at low and room temperature the nanobeams’ transverse deflection decreased with increase in temperature
difference, whereas at higher temperature the transverse deflections increased as the
temperature difference increased.
Chen et al. [21] predicted the damping behaviour of the nanobeams including
thermal fluctuations and the paddling effect.