7.7 Mathematical Model of Three-Layer Micro- and Nano-Beams
265
Thai and Vo [188] studied static bending, buckling and free vibration behaviours
of size-dependent functionally graded sandwich microbeams based on the modified
couple stress theory and Timoshenko beam theory. Two kinds of the sandwich beams
have been analysed: functionally graded skins and homogeneous core and functionally graded core and homogeneous skins. It has been shown that inclusion of the size
effect resulted in an increase in the beam stiffness.
In Ref. [189], a size-dependent formulation for Bernoulli-Euler beam based on
the couple stress theory has been given. It has been shown that the natural frequencies
obtained using the utilized couple stress model are higher than these predicted by
the classical theory.
A new modified couple stress theory containing three material length scale
parameters has been developed by Chen and Li [190] for anisotropic elasticity and
microscale laminated Kirchhoff plate model. The principle of minimum total potential energy has been employed, and the curvature tensor has been taken as asymmetric,
whereas the couple stress moment tensor has been used as symmetric. The carried
out numerical simulation have validated the proposed Kirchhoff plate model, which
captures the scale effect of the microstructures.
Mohammad-Abadi and Daneshmehr [191] carried out the vibration analysis of the
composite laminated beams in order of micron based on the modified couple stress
theory. In particular, Euler-Bernoulli, Timoshenko and Reddy beam models have
been studied with respect to the differences in the estimation of shear deformation.
The governing equations have been solved using three-boundary conditions and four
types of lamination.
The transverse vibration of rotary tapered microbeam has been analysed by Shafiei
et al. [192] using a modified couple stress theory and Euler-Bernoulli beam model. In
particular, the effect of the small-scale parameter, beam length, rate of cross-sectional
change, hub radius and non-dimensional angular velocity on the microbeam vibration
process have been illustrated and discussed.
In Ref. [72], the size-dependent model of a three-layer beam employing the modified version of the couple stress theory and length parameter influence associated with
all layers and its impact on the damping characteristics of the microbeam vibrations
have been studied.
Rajneesh [193] solved the problem of thermoelastic beam using the modified
couple stress theory. Both governing equations for the modified couple stress theory
and heat conduction equation for coupled thermoelasticity have been employed to
study the vibrations in a homogenous isotropic thin beam by applying Euler-Bernoulli
theory. The lateral deflection, thermal moment, axial stress average due to normal
heat flux have been derived and studied numerically.
In Ref. [194], shear deformable functionally graded nanobeams in post-buckling
based on modified couple stress theory have been studied. The governing equations
and boundary conditions are yielded by the principle of minimum potential energy.
Exact and generalized differential quadrature solutions for the static post-buckling
response of the functionally graded nanobeams under different boundary conditions
have been derived. Effects of length scale parameter, material gradient, length-tothickness ratio and Poisson’s ratios have been illustrated and analysed, among others.
265
Thai and Vo [188] studied static bending, buckling and free vibration behaviours
of size-dependent functionally graded sandwich microbeams based on the modified
couple stress theory and Timoshenko beam theory. Two kinds of the sandwich beams
have been analysed: functionally graded skins and homogeneous core and functionally graded core and homogeneous skins. It has been shown that inclusion of the size
effect resulted in an increase in the beam stiffness.
In Ref. [189], a size-dependent formulation for Bernoulli-Euler beam based on
the couple stress theory has been given. It has been shown that the natural frequencies
obtained using the utilized couple stress model are higher than these predicted by
the classical theory.
A new modified couple stress theory containing three material length scale
parameters has been developed by Chen and Li [190] for anisotropic elasticity and
microscale laminated Kirchhoff plate model. The principle of minimum total potential energy has been employed, and the curvature tensor has been taken as asymmetric,
whereas the couple stress moment tensor has been used as symmetric. The carried
out numerical simulation have validated the proposed Kirchhoff plate model, which
captures the scale effect of the microstructures.
Mohammad-Abadi and Daneshmehr [191] carried out the vibration analysis of the
composite laminated beams in order of micron based on the modified couple stress
theory. In particular, Euler-Bernoulli, Timoshenko and Reddy beam models have
been studied with respect to the differences in the estimation of shear deformation.
The governing equations have been solved using three-boundary conditions and four
types of lamination.
The transverse vibration of rotary tapered microbeam has been analysed by Shafiei
et al. [192] using a modified couple stress theory and Euler-Bernoulli beam model. In
particular, the effect of the small-scale parameter, beam length, rate of cross-sectional
change, hub radius and non-dimensional angular velocity on the microbeam vibration
process have been illustrated and discussed.
In Ref. [72], the size-dependent model of a three-layer beam employing the modified version of the couple stress theory and length parameter influence associated with
all layers and its impact on the damping characteristics of the microbeam vibrations
have been studied.
Rajneesh [193] solved the problem of thermoelastic beam using the modified
couple stress theory. Both governing equations for the modified couple stress theory
and heat conduction equation for coupled thermoelasticity have been employed to
study the vibrations in a homogenous isotropic thin beam by applying Euler-Bernoulli
theory. The lateral deflection, thermal moment, axial stress average due to normal
heat flux have been derived and studied numerically.
In Ref. [194], shear deformable functionally graded nanobeams in post-buckling
based on modified couple stress theory have been studied. The governing equations
and boundary conditions are yielded by the principle of minimum potential energy.
Exact and generalized differential quadrature solutions for the static post-buckling
response of the functionally graded nanobeams under different boundary conditions
have been derived. Effects of length scale parameter, material gradient, length-tothickness ratio and Poisson’s ratios have been illustrated and analysed, among others.