2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
43
the nanobeam for two types of the transverse cross sections were obtained. Their
dependence on the surface relaxation and surface tension were reported.
Yet, there was no theoretical model created which could adequately explain all
phenomena of the size dependence. Villain et al. [229] detected a strong decrease
of Young modulus with the help of atomic modelling with account of the surface
tension. Employing the method of molecular statics based on the atomic potential
Wolf [230] and Liang [231] showed that the modulus of elasticity can increase or
decrease on the nano-level. The similar results were also obtained by Zhou and Huang
[232] with the use of combinations of the molecular statics theory and computations.
Miller and Shenoy [37] developed the model consisting of surface tension in order
to forecast elastic properties of the nanobeams and nanoplates. They concluded that
the effective stiffness is closely related to the values of elastic surface constants.
Streitz and Cammarata [233] analysed the influence of surface energy on the elastic
properties of the nanomaterials and forecasted increase of the elastic modulus of
nanomembranes. Dingreville et al. [234] worked out a structural scheme in order
to include the surface energy in the theory of continuum and demonstrated how
the general model of elasticity of nanostructures can either increase or decrease.
Guo and Zhao [235] also developed a 3D model with a layer composed of relaxed
atoms which served as a basis of investigation of size-dependent elastic constants
of nanomembranes and showed that moduli of elasticity of a nanomembrane may
either increase or decrease while decreasing of its thickness.
Non-classical model of Mindlin plate located on elastic foundation was studied with the help of modified theory of couple stresses, theory of surface elasticity
and two-parametric model of the Winkler-Pasternak foundation [236]. The model
includes five kinematic parameters, and the governing equations and boundary conditions were yielded by the Hamilton principle. Microstructure effects, surface energy
and fundamental effects are taken into account. It included one material scale parameter useful for descriptions of the microstructure effects, three elasticity constants of
the surface needed to include surface energy and two parameters of foundation effect.
The derived non-classical model can be reduced to its classical counterpart form
based on elasticity theory assuming that the microstructure effects, surface energy
and foundation effects are removed. Besides, this new model includes the Mindlin
plate model which takes into account microstructural effects, surface energy influence and influence of the foundation being treated as separate cases in the carried
out investigations.
Thin beams and plates supported by elastic foundation are widely employed in
nano- and microscale devices and systems. There were presented various types of
the elastic foundation models including those derived by Winkler [237], FilonenkoBorodich [238], Pasternak [239], Kerr [240] and Vlasov [241]. Winkler’s model consists of only one parameter and it is the simplest one, whereas the Winkler-Pasternak
model consists of two parameters and may exhibit the influence of foundation effects
[242–244]. Since the size effects play an important role in applications in the nanoand microscales while the classical continuum theories cannot demonstrate the latter
effects, more recently continuum theories of higher orders were worked out. The latter
includes the parameters of material length and may include effects of microstructure
43
the nanobeam for two types of the transverse cross sections were obtained. Their
dependence on the surface relaxation and surface tension were reported.
Yet, there was no theoretical model created which could adequately explain all
phenomena of the size dependence. Villain et al. [229] detected a strong decrease
of Young modulus with the help of atomic modelling with account of the surface
tension. Employing the method of molecular statics based on the atomic potential
Wolf [230] and Liang [231] showed that the modulus of elasticity can increase or
decrease on the nano-level. The similar results were also obtained by Zhou and Huang
[232] with the use of combinations of the molecular statics theory and computations.
Miller and Shenoy [37] developed the model consisting of surface tension in order
to forecast elastic properties of the nanobeams and nanoplates. They concluded that
the effective stiffness is closely related to the values of elastic surface constants.
Streitz and Cammarata [233] analysed the influence of surface energy on the elastic
properties of the nanomaterials and forecasted increase of the elastic modulus of
nanomembranes. Dingreville et al. [234] worked out a structural scheme in order
to include the surface energy in the theory of continuum and demonstrated how
the general model of elasticity of nanostructures can either increase or decrease.
Guo and Zhao [235] also developed a 3D model with a layer composed of relaxed
atoms which served as a basis of investigation of size-dependent elastic constants
of nanomembranes and showed that moduli of elasticity of a nanomembrane may
either increase or decrease while decreasing of its thickness.
Non-classical model of Mindlin plate located on elastic foundation was studied with the help of modified theory of couple stresses, theory of surface elasticity
and two-parametric model of the Winkler-Pasternak foundation [236]. The model
includes five kinematic parameters, and the governing equations and boundary conditions were yielded by the Hamilton principle. Microstructure effects, surface energy
and fundamental effects are taken into account. It included one material scale parameter useful for descriptions of the microstructure effects, three elasticity constants of
the surface needed to include surface energy and two parameters of foundation effect.
The derived non-classical model can be reduced to its classical counterpart form
based on elasticity theory assuming that the microstructure effects, surface energy
and foundation effects are removed. Besides, this new model includes the Mindlin
plate model which takes into account microstructural effects, surface energy influence and influence of the foundation being treated as separate cases in the carried
out investigations.
Thin beams and plates supported by elastic foundation are widely employed in
nano- and microscale devices and systems. There were presented various types of
the elastic foundation models including those derived by Winkler [237], FilonenkoBorodich [238], Pasternak [239], Kerr [240] and Vlasov [241]. Winkler’s model consists of only one parameter and it is the simplest one, whereas the Winkler-Pasternak
model consists of two parameters and may exhibit the influence of foundation effects
[242–244]. Since the size effects play an important role in applications in the nanoand microscales while the classical continuum theories cannot demonstrate the latter
effects, more recently continuum theories of higher orders were worked out. The latter
includes the parameters of material length and may include effects of microstructure