2.2 Literature Review
27
named as the nonlocal theory of deformations gradient has been developed. It is
based on unification of the theory of elasticity and theory of deformations gradient
(see [34–36]).
Though a big achievement has been obtained for the magneto-electro-elastic material (MEU) with microscopic dimension, the application of the traditional theory of
elasticity being based on microscopic characteristics does not allow to achieve reliable results in the nanoscale. Namely, the influence of size-dependent effects on the
mechanical properties of materials is observed on the nanosize level what has been
confirmed by atomic modelling and experiments [37–41]. In the nanoscale, a surface
has remarkable influence on the general size-dependent mechanical properties of
materials. From the mathematical point of view, the materials can be presented as a
surface layer with zero-order thickness and volume kernel with mechanical properties being different from the surface layer. Based on the surface theory of elasticity,
many different important problems have been investigated including influence of the
surface on the bending [42–44], stability [45–47] and propagation of waves [48–
51] as well as free vibrations [47, 52–60] of nanomaterials and nanostructures. The
mentioned works pointed out the importance of account of the surface elasticity and
surface stress into the fundamental governing equation for getting structural characteristics of nanomaterials. Based on the theory of surface energy, it is assumed that
the surface properties cannot be ignored while studying nanostructures and nanomaterials due to large ratio of the surface area and the amount of volume in the
nanosize structures [61]. The size-dependent behaviour in nanostructures has been
investigated in [43, 62].
The size-dependent models are widely employed for prognosis of global behaviour
of beam, plate and shell nanostructures such as carbon nanotubes (CNTs) and
graphene leaves. CNTs have been first discovered by Iijima [63]. Depending on
the fabrication technology, one may produce various types of CNTs, such as singlewall nanotubes (SWCNTs), double-wall nanotubes (DWCNTs) and multi-wall nanotubes (MWCNTs). They can be obtained by twisting of single-layer graphene sheets
(SLGS), double-layer graphene sheets (DLGS) and multi-layer graphene sheets
(MLGS) in all areas of nanotechnology [64–70].
One of the important aspects of the use of the size-dependent models relies on
their application to the problems of statics and dynamics of beams. There is now
available a palette of various size-dependent models in theory of beams and plates.
The simplest models are based on theory of Euler-Bernoulli (EBT) and the classical
theory of plates (CPT). Those models are applicable only for modelling of thin
beams and thin plates, since they do not include effects of shear deformation. In
order to overcome limitation of EBT and CPT, a series of theories according to
shear deformation have been proposed. Models including shear deformations of the
first order are based on the theory of Timoshenko beams (TBT) and theory of shear
deformation of the first order (FSDT). Since in those models plane displacements
undergo changes along thickness, there is a need to introduce correction coefficient
of the shear effect. In order to remove the mentioned correction coefficient and to
get more exact estimation of deflection in thick beams and plates, a few theories
of higher order shear deformation (HSDT) are proposed including Reddy’s beam
27
named as the nonlocal theory of deformations gradient has been developed. It is
based on unification of the theory of elasticity and theory of deformations gradient
(see [34–36]).
Though a big achievement has been obtained for the magneto-electro-elastic material (MEU) with microscopic dimension, the application of the traditional theory of
elasticity being based on microscopic characteristics does not allow to achieve reliable results in the nanoscale. Namely, the influence of size-dependent effects on the
mechanical properties of materials is observed on the nanosize level what has been
confirmed by atomic modelling and experiments [37–41]. In the nanoscale, a surface
has remarkable influence on the general size-dependent mechanical properties of
materials. From the mathematical point of view, the materials can be presented as a
surface layer with zero-order thickness and volume kernel with mechanical properties being different from the surface layer. Based on the surface theory of elasticity,
many different important problems have been investigated including influence of the
surface on the bending [42–44], stability [45–47] and propagation of waves [48–
51] as well as free vibrations [47, 52–60] of nanomaterials and nanostructures. The
mentioned works pointed out the importance of account of the surface elasticity and
surface stress into the fundamental governing equation for getting structural characteristics of nanomaterials. Based on the theory of surface energy, it is assumed that
the surface properties cannot be ignored while studying nanostructures and nanomaterials due to large ratio of the surface area and the amount of volume in the
nanosize structures [61]. The size-dependent behaviour in nanostructures has been
investigated in [43, 62].
The size-dependent models are widely employed for prognosis of global behaviour
of beam, plate and shell nanostructures such as carbon nanotubes (CNTs) and
graphene leaves. CNTs have been first discovered by Iijima [63]. Depending on
the fabrication technology, one may produce various types of CNTs, such as singlewall nanotubes (SWCNTs), double-wall nanotubes (DWCNTs) and multi-wall nanotubes (MWCNTs). They can be obtained by twisting of single-layer graphene sheets
(SLGS), double-layer graphene sheets (DLGS) and multi-layer graphene sheets
(MLGS) in all areas of nanotechnology [64–70].
One of the important aspects of the use of the size-dependent models relies on
their application to the problems of statics and dynamics of beams. There is now
available a palette of various size-dependent models in theory of beams and plates.
The simplest models are based on theory of Euler-Bernoulli (EBT) and the classical
theory of plates (CPT). Those models are applicable only for modelling of thin
beams and thin plates, since they do not include effects of shear deformation. In
order to overcome limitation of EBT and CPT, a series of theories according to
shear deformation have been proposed. Models including shear deformations of the
first order are based on the theory of Timoshenko beams (TBT) and theory of shear
deformation of the first order (FSDT). Since in those models plane displacements
undergo changes along thickness, there is a need to introduce correction coefficient
of the shear effect. In order to remove the mentioned correction coefficient and to
get more exact estimation of deflection in thick beams and plates, a few theories
of higher order shear deformation (HSDT) are proposed including Reddy’s beam