42
2 Size-Dependent Theories of Beams, Plates and Shells
beam [219]. The surface effects are taken into account in the modified theory of beams
via the model regarding the surface piezoelectricity and the generalized YoungLaplace equation, whereas flexoelectricity effects are considered with the help of
theory of piezoelectricity of higher order. The results of modelling of dynamics of
piezoelectric nanobeams show that the influence of surface effects and flexoelectricity effects are changed with respect to the beam thickness and beam size ratio.
It should be emphasized that due to large ratio of the surface area to volume of
typical nanostructures, the size effects are coupled with surface effects. Gurtin and
Merdok [61] proposed the model of surface elasticity for elastic materials, where the
surface is modelled as thin layer with small thickness attached to the volume. The
governing and equilibrium equations for the surface layer differ from those inside the
volume of the rigid body. Using that model, the size-dependent properties of nanomaterials were reported in [42, 46, 216]. However, the model of surface elasticity
is not sufficient for investigation of mechanical behaviour of piezoelectric nanostructure with electromechanical couplings. In order to solve these problems for the
piezoelectric nanostructures, the modified continuum model with surface-dependent
effects depended on electric field was developed in [220]. In that model, the surface
stresses depended on surface piezoelectricity and played a role of supplement to the
surface elasticity. In that new surface piezoelectric model, both static and dynamic
behaviours of various piezoelectric nanostructures were investigated [220–223]. The
modelling results showed that the surface effects exhibit essential influence on properties of the static bending, vibrations and stability of the mentioned piezoelectric
nanostructures.
It is also recognized that the flexoelectricity is responsible for size-dependent
properties of piezoelectric nanomaterials which concerns a spontaneous polarization of electric materials due to either deformation gradient or non-homogeneous
deformation field. Maranganti et al. [224] proposed a theoretical basis for dielectrics
with an account of flexoelectricity and clarified mechanism of electromechanical
coupling owing to gradients of deformations and/or polarization effects. Majdoub et
al. [225] demonstrated strong increase of the effective piezoelectric coefficients on
the size which depends on the size and included the flexoelectric phenomena into
theory of piezoelectricity. Eliseev et al. [226] investigated renormalization process in
properties of nanoferroids due to spontaneous flexoelectric effect. More recently, Lu
et al. [227] investigated the influence of flexoelectricity on the electrostatic potential
in the piezoelectric nanotube. The described so far investigations imply necessity
of the account of flexoelectric effect while characterizing properties of piezoelectric
nanostructures.
In Ref. [228], the theoretical model for investigation of the size-dependent elastic
properties of nanobeam with an account of surface relaxation effects and surface tension was studied. The surface layer and its thickness were estimated in a unique way.
The 3D model of nanofilm with a relaxation layer of atoms was analysed. Four nonzero elastic constants of the nanomembrane were obtained, and then Young modulus
for straight extension was derived. Employing the ratio of energetic equilibrium,
the dependences of effective elasticity modulus and effective bending stiffness of
2 Size-Dependent Theories of Beams, Plates and Shells
beam [219]. The surface effects are taken into account in the modified theory of beams
via the model regarding the surface piezoelectricity and the generalized YoungLaplace equation, whereas flexoelectricity effects are considered with the help of
theory of piezoelectricity of higher order. The results of modelling of dynamics of
piezoelectric nanobeams show that the influence of surface effects and flexoelectricity effects are changed with respect to the beam thickness and beam size ratio.
It should be emphasized that due to large ratio of the surface area to volume of
typical nanostructures, the size effects are coupled with surface effects. Gurtin and
Merdok [61] proposed the model of surface elasticity for elastic materials, where the
surface is modelled as thin layer with small thickness attached to the volume. The
governing and equilibrium equations for the surface layer differ from those inside the
volume of the rigid body. Using that model, the size-dependent properties of nanomaterials were reported in [42, 46, 216]. However, the model of surface elasticity
is not sufficient for investigation of mechanical behaviour of piezoelectric nanostructure with electromechanical couplings. In order to solve these problems for the
piezoelectric nanostructures, the modified continuum model with surface-dependent
effects depended on electric field was developed in [220]. In that model, the surface
stresses depended on surface piezoelectricity and played a role of supplement to the
surface elasticity. In that new surface piezoelectric model, both static and dynamic
behaviours of various piezoelectric nanostructures were investigated [220–223]. The
modelling results showed that the surface effects exhibit essential influence on properties of the static bending, vibrations and stability of the mentioned piezoelectric
nanostructures.
It is also recognized that the flexoelectricity is responsible for size-dependent
properties of piezoelectric nanomaterials which concerns a spontaneous polarization of electric materials due to either deformation gradient or non-homogeneous
deformation field. Maranganti et al. [224] proposed a theoretical basis for dielectrics
with an account of flexoelectricity and clarified mechanism of electromechanical
coupling owing to gradients of deformations and/or polarization effects. Majdoub et
al. [225] demonstrated strong increase of the effective piezoelectric coefficients on
the size which depends on the size and included the flexoelectric phenomena into
theory of piezoelectricity. Eliseev et al. [226] investigated renormalization process in
properties of nanoferroids due to spontaneous flexoelectric effect. More recently, Lu
et al. [227] investigated the influence of flexoelectricity on the electrostatic potential
in the piezoelectric nanotube. The described so far investigations imply necessity
of the account of flexoelectric effect while characterizing properties of piezoelectric
nanostructures.
In Ref. [228], the theoretical model for investigation of the size-dependent elastic
properties of nanobeam with an account of surface relaxation effects and surface tension was studied. The surface layer and its thickness were estimated in a unique way.
The 3D model of nanofilm with a relaxation layer of atoms was analysed. Four nonzero elastic constants of the nanomembrane were obtained, and then Young modulus
for straight extension was derived. Employing the ratio of energetic equilibrium,
the dependences of effective elasticity modulus and effective bending stiffness of