2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
41
[209], Allahbakhshi and Allahbakhshi [210], Li et al. [211], Hosseini et al. [212] and
Zhang et al. [213] extended the CPT model based on theory of deformations gradient
into SWCNT [209], MLGS [210], two-layer isotropic microplates [211], multi-layer
orthotropic microplates [212] and isotropic microplates lying on elastic basis [213].
2.3.5.4 Beams (EBT), Plates (CPT) and Shells (KLT) Models Based on
the Surface Theory of Elasticity
In order to achieve high accuracy models of size-dependent behaviour in nanostructures, both surface and volume effects are taken into account. Theory of surface energy and theory of elasticity of deformation gradient are introduced to fit
both surface and volume effects, respectively. There are proposed novel models for
the Bernoulli-Euler and Timoshenko beams. The fundamental equations, initial and
boundary conditions are introduced simultaneously using the Hamilton principle.
New models include the Poisson’s effect and have three parameters of material length
and three constants of the surface elasticity to the account of scale material length as
well as three constants of the surface elasticity in order to quantify size effects in the
surface and volume of the nanostructure, as well as the modified couple stress models
and classical models in order to illustrate features of novel models regarding the problems of static bending and free vibrations of simply supported Euler-Bernoulli and
Timoshenko nanobeams. Numerical results show that the differences in deflections
and eigenfrequency predicted by the mentioned model and other models are large for
small beam thickness. The differences are decreased when the beam size increases.
There are considered Euler-Bernoulli and Timoshenko models based on the continuum theory of Gurtin-Merdok to study thin and thick nanobeams with arbitrary
cross section [214]. Gao et al. [60, 215, 216] also proposed models of beams and
plates including the size effect of the surface energy while studying size-dependent
mechanical properties. Theory of surface energy characterizing the surface effect
takes into account only the influence of surface layer. The modified couple stress
theory and theory of deformations gradient characterizing the size effect take into
account the influence of volume material. In other words, the mentioned theories
characterize the size effect either for the volume part or for the surface.
In the reviewed literature, majority of the works characterize the size effect in
nanostructures either from the point of view of volume or surface [206, 217]. There
is a limited number of works to study size effect including both features [60, 215].
In Ref. [218], there is proposed a universal beam model taking into account both
surface theory of elasticity and the volume theory. There are obtained formulation
of nanosized Bernoulli-Euler and Timoshenko beams under the theory of gradient
elasticity and theory of surface energy. They yielded the governing equations, initial
conditions and all possible boundary conditions. There are solved problems of static
bending and free vibrations of Bernoulli-Euler and Timoshenko beams with simply
supported ends.
The size-dependent characteristics of piezoelectric nanomaterials which are coupled with surface effects and flexoelectricity are investigated for the Euler-Bernoulli
41
[209], Allahbakhshi and Allahbakhshi [210], Li et al. [211], Hosseini et al. [212] and
Zhang et al. [213] extended the CPT model based on theory of deformations gradient
into SWCNT [209], MLGS [210], two-layer isotropic microplates [211], multi-layer
orthotropic microplates [212] and isotropic microplates lying on elastic basis [213].
2.3.5.4 Beams (EBT), Plates (CPT) and Shells (KLT) Models Based on
the Surface Theory of Elasticity
In order to achieve high accuracy models of size-dependent behaviour in nanostructures, both surface and volume effects are taken into account. Theory of surface energy and theory of elasticity of deformation gradient are introduced to fit
both surface and volume effects, respectively. There are proposed novel models for
the Bernoulli-Euler and Timoshenko beams. The fundamental equations, initial and
boundary conditions are introduced simultaneously using the Hamilton principle.
New models include the Poisson’s effect and have three parameters of material length
and three constants of the surface elasticity to the account of scale material length as
well as three constants of the surface elasticity in order to quantify size effects in the
surface and volume of the nanostructure, as well as the modified couple stress models
and classical models in order to illustrate features of novel models regarding the problems of static bending and free vibrations of simply supported Euler-Bernoulli and
Timoshenko nanobeams. Numerical results show that the differences in deflections
and eigenfrequency predicted by the mentioned model and other models are large for
small beam thickness. The differences are decreased when the beam size increases.
There are considered Euler-Bernoulli and Timoshenko models based on the continuum theory of Gurtin-Merdok to study thin and thick nanobeams with arbitrary
cross section [214]. Gao et al. [60, 215, 216] also proposed models of beams and
plates including the size effect of the surface energy while studying size-dependent
mechanical properties. Theory of surface energy characterizing the surface effect
takes into account only the influence of surface layer. The modified couple stress
theory and theory of deformations gradient characterizing the size effect take into
account the influence of volume material. In other words, the mentioned theories
characterize the size effect either for the volume part or for the surface.
In the reviewed literature, majority of the works characterize the size effect in
nanostructures either from the point of view of volume or surface [206, 217]. There
is a limited number of works to study size effect including both features [60, 215].
In Ref. [218], there is proposed a universal beam model taking into account both
surface theory of elasticity and the volume theory. There are obtained formulation
of nanosized Bernoulli-Euler and Timoshenko beams under the theory of gradient
elasticity and theory of surface energy. They yielded the governing equations, initial
conditions and all possible boundary conditions. There are solved problems of static
bending and free vibrations of Bernoulli-Euler and Timoshenko beams with simply
supported ends.
The size-dependent characteristics of piezoelectric nanomaterials which are coupled with surface effects and flexoelectricity are investigated for the Euler-Bernoulli