2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
51
but it was spanned on four FSDT unknowns proposed by Thai et al. [355–357].
Therefore, their model was simpler than that proposed by Gholami et al. [348],
which included five unknowns.
2.3.6.4 Surface Theory of Timoshenko
A complex Timoshenko beam model to study static bending problems of nanowires/
nanobeams with surface effects was used in [43]. There were obtained explicit solutions of the nonlinear problem with various boundary conditions for investigation
of the combined influence of the residual surface stress, surface elasticity and shear
deformation on the effective stiffness and Young modulus. It was shown that stiffness
depends on the size, which is important for thin beams. The residual surface stress has
a tendency to increase or decrease stiffness of a nanobeam versus boundary conditions. Shear deformation always makes the nanobeam more soft in comparison with
the Euler-Bernoulli model. The obtained solutions well coincided with the experimental measurement of the Young modulus estimation, in particular, for a beam with
small ratio of its length to thickness.
2.3.7 Size-Dependent Theory of Beams, Plates and Shells
Based on the Shear Deformation Model of the Third
Order
2.3.7.1 Nonlocal Theory of RBT, TSDT, HSDT
Based on the nonlocal constitutive relations, Reddy [358] modified theory of beams
(EBT, TBT, RBT) in order to take into account a nonlocal effect. There were obtained
variational equations for four models which were used to account of nonlocal models
based on finite elements. For the case of simplified beams, there were obtained solutions in closed forms of the deflections, longitudinal loads and the eigenfrequencies.
Ebrahimi and Salari [359] introduced thermal effects into nonlocal RBT to study
the influence of temperature and nonlocal parameter on the characteristics of free
vibration of OCNT. Emam [360] proposed a unified nonlinear nonlocal model for
analysis of stability of isotropic nanobeams. There were obtained analytical solutions
for critical load estimation for simply supported and clamped nanobeams.
Rahmani and Jandaghian [361] extended the nonlocal RBT model into the FG
nanobeams. Analytical solutions for the critical loads were obtained for the FG
nanobeams for various boundary conditions with the use of the Rayleigh-Ritz
method. Ebrahimi and Reza Barati [362] also worked out the nonlocal RBT model
for FG nanobeams where thermal effects and interactions between nanobeam and
elastic medium were considered.
51
but it was spanned on four FSDT unknowns proposed by Thai et al. [355–357].
Therefore, their model was simpler than that proposed by Gholami et al. [348],
which included five unknowns.
2.3.6.4 Surface Theory of Timoshenko
A complex Timoshenko beam model to study static bending problems of nanowires/
nanobeams with surface effects was used in [43]. There were obtained explicit solutions of the nonlinear problem with various boundary conditions for investigation
of the combined influence of the residual surface stress, surface elasticity and shear
deformation on the effective stiffness and Young modulus. It was shown that stiffness
depends on the size, which is important for thin beams. The residual surface stress has
a tendency to increase or decrease stiffness of a nanobeam versus boundary conditions. Shear deformation always makes the nanobeam more soft in comparison with
the Euler-Bernoulli model. The obtained solutions well coincided with the experimental measurement of the Young modulus estimation, in particular, for a beam with
small ratio of its length to thickness.
2.3.7 Size-Dependent Theory of Beams, Plates and Shells
Based on the Shear Deformation Model of the Third
Order
2.3.7.1 Nonlocal Theory of RBT, TSDT, HSDT
Based on the nonlocal constitutive relations, Reddy [358] modified theory of beams
(EBT, TBT, RBT) in order to take into account a nonlocal effect. There were obtained
variational equations for four models which were used to account of nonlocal models
based on finite elements. For the case of simplified beams, there were obtained solutions in closed forms of the deflections, longitudinal loads and the eigenfrequencies.
Ebrahimi and Salari [359] introduced thermal effects into nonlocal RBT to study
the influence of temperature and nonlocal parameter on the characteristics of free
vibration of OCNT. Emam [360] proposed a unified nonlinear nonlocal model for
analysis of stability of isotropic nanobeams. There were obtained analytical solutions
for critical load estimation for simply supported and clamped nanobeams.
Rahmani and Jandaghian [361] extended the nonlocal RBT model into the FG
nanobeams. Analytical solutions for the critical loads were obtained for the FG
nanobeams for various boundary conditions with the use of the Rayleigh-Ritz
method. Ebrahimi and Reza Barati [362] also worked out the nonlocal RBT model
for FG nanobeams where thermal effects and interactions between nanobeam and
elastic medium were considered.