2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
45
for functionally graded membrane with the help of Kirchhoff’s kinematic relations
and nonlinear von Kármán deformation of a volume material. Wang and Wang [253]
proposed the model of nonlinear vibrations of the Kirchhoff plate and a Mindlin plate
with the help of von Kármán model. However, in general, there is a small amount
of models for thin plates with an account of both microstructure and surface energy.
One of the non-classical models of thin Kirchhoff’s plates using theory of elasticity of deformation gradient was proposed in [254]. It consisted of two additional
scale length parameters: one was associated with energy volume deformation and
the second represented energy of surface [216].
Influence of the surface stresses on rigidity of the cantilever plates within the
employment of 3D model was investigated in [255]. A link between surface stress
and stiffness of the plate was proposed with an account of the plate sizes. However, in
general, the use of the short nanosized plates stands for the most challenging future
investigations.
Rouhi et al. [256] proposed an analytical approach for analysis of geometrically
nonlinear free vibrations of cylindrical nanoshells. The continuum model of GurtinMerdok was employed for the account of surface stresses. The governing PDEs
of the shell with the surface stresses were derived based on the energetic method.
The amplitude-frequency characteristics were reported. There were given numerous
numerical results for the investigation of dynamic behaviour of nanosized shells with
various geometric and surface material properties. It was shown that the surface stress
depends essentially on the behaviour of nonlinear free vibrations of nanosized shells
being very thin. Besides, it was detected that the effect of geometric nonlinearity was
more exhibited when the residual stress was negative.
2.3.6 Size-Dependent Theory of Beams, Plates and Shells
Based on the Shear Deformations of the First Order
Under the Timoshenko Theory (TBT)
2.3.6.1 Nonlocal Timoshenko Models
The earliest nonlocal TBT model was proposed by Wang [257] to study wave propagation in CNT. The latter models account for the shear deformation which becomes
essential for short and thick CNT. Wang and Varadan [258] also developed the
nonlocal TBT model which was employed for the investigation of free vibrations
either as SWCNTs or DWCNTs. The closed-form solutions for eigenfrequencies
of simply supported CNT were reported. Wang et al. [259–262] obtained solutions
in closed form for the critical loads [259], eigenfrequencies [260] and deflections
[261, 262] with the help of nonlocal TBT model with four different boundary conditions including simply supported clamped cantilever and propped cantilever. In those
models, the transversal shear stress was estimated based on the local theory. Lu et
al. [263] included the nonlocal effect into both normal and transverse shear effects.
45
for functionally graded membrane with the help of Kirchhoff’s kinematic relations
and nonlinear von Kármán deformation of a volume material. Wang and Wang [253]
proposed the model of nonlinear vibrations of the Kirchhoff plate and a Mindlin plate
with the help of von Kármán model. However, in general, there is a small amount
of models for thin plates with an account of both microstructure and surface energy.
One of the non-classical models of thin Kirchhoff’s plates using theory of elasticity of deformation gradient was proposed in [254]. It consisted of two additional
scale length parameters: one was associated with energy volume deformation and
the second represented energy of surface [216].
Influence of the surface stresses on rigidity of the cantilever plates within the
employment of 3D model was investigated in [255]. A link between surface stress
and stiffness of the plate was proposed with an account of the plate sizes. However, in
general, the use of the short nanosized plates stands for the most challenging future
investigations.
Rouhi et al. [256] proposed an analytical approach for analysis of geometrically
nonlinear free vibrations of cylindrical nanoshells. The continuum model of GurtinMerdok was employed for the account of surface stresses. The governing PDEs
of the shell with the surface stresses were derived based on the energetic method.
The amplitude-frequency characteristics were reported. There were given numerous
numerical results for the investigation of dynamic behaviour of nanosized shells with
various geometric and surface material properties. It was shown that the surface stress
depends essentially on the behaviour of nonlinear free vibrations of nanosized shells
being very thin. Besides, it was detected that the effect of geometric nonlinearity was
more exhibited when the residual stress was negative.
2.3.6 Size-Dependent Theory of Beams, Plates and Shells
Based on the Shear Deformations of the First Order
Under the Timoshenko Theory (TBT)
2.3.6.1 Nonlocal Timoshenko Models
The earliest nonlocal TBT model was proposed by Wang [257] to study wave propagation in CNT. The latter models account for the shear deformation which becomes
essential for short and thick CNT. Wang and Varadan [258] also developed the
nonlocal TBT model which was employed for the investigation of free vibrations
either as SWCNTs or DWCNTs. The closed-form solutions for eigenfrequencies
of simply supported CNT were reported. Wang et al. [259–262] obtained solutions
in closed form for the critical loads [259], eigenfrequencies [260] and deflections
[261, 262] with the help of nonlocal TBT model with four different boundary conditions including simply supported clamped cantilever and propped cantilever. In those
models, the transversal shear stress was estimated based on the local theory. Lu et
al. [263] included the nonlocal effect into both normal and transverse shear effects.