44
2 Size-Dependent Theories of Beams, Plates and Shells
and surface energy. Khajeansari et al. [244] investigated the bending deformation
of the Euler-Bernoulli beam lying on the Winkler-Pasternak foundation employing
theory of surface elasticity (see, for instance, [94, 245]). Simsek and Reddy [246]
proposed the functionally gradient model of a microbeam embedded into an elastic
medium using the modified couple stress theory (see [15, 247]) and the WinklerPasternak foundation. Limkatanyu et al. [248] proposed the Euler-Bernoulli beam
model supported by the Winkler-Pasternak foundation and included the microstructure and surface energy effects, which extended the non-classical model proposed
by Gao and Mahmuda [218]. More recently, Gao and Zhang [249] proposed the
non-classical model which takes into account the microstructure, surface energy
and influence of the foundation for the Kirchhoff plates with the use of variational
formulation.
Novel non-classical model of Kirchhoff plates with the use of the modified couple
stress theory, the surface theory of elasticity and two-parameter model of elastic foundation was presented in [249]. The variational formulation based on the Hamilton
principle which yielded the derived equations of motion and the boundary conditions was used. New plate model consisted of the scale material length parameter to
account of the microstructure effect, three elastic surface constants for description
of the surface energy, and two elastic moduli of foundation. In order to demonstrate
model abilities, the problem of static bending and free vibrations of the rectangular
plate simply supported was solved analytically. In the case of the static bending, the
numerical results indicate that in the considered cases deflection of the plate lying on
the elastic foundation is less than that forecasted by the classical models. Besides,
a difference in deflection forecast by new and classical models of plates is of large
amount if the plate thickness is efficiently small (it is decreased with increase of a
plate thickness). In the case of free vibrations, it was found that the eigenfrequencies
obtained for new plate models under elastic foundation or without are higher than
those obtained by the classical plate models. The so far described size effects in
microscale are validated by laboratory experiments. Besides, it was shown both analytically and numerically that an occurrence of elastic foundation implies decrease
of a plate deflection and increase of fundamental plate frequency, what was to be
expected. In the case of a rigid body with large ratio of the surface to its volume, the
size effects play a crucial role [37]. The size effects can be understood with the help
of theory of surface elasticity where locations of the atoms and material properties
are different in the surface and inside the body volume, and one surface effect is
included into artificially constructed member with a negligible thickness [87, 95,
96].
Theory of surface elasticity [61, 94] was employed for analysis of thin plates coupled with surface effects. For instance, Miller and Shenoy [37] developed a model
suitable for the description of the dependence of effective stiffness of nanosized
structural element (rod, beam or plate) on the size. Lim and He [250] proposed a
geometrically nonlinear model of nanoplates and nanomembranes based on Kirchhoff’s hypothesis and von Kármán deformations. Lu et al. [251] developed the model
of a thin plate dependent on the size with inclusion of normal stresses out and inside
of the surface of volume layer. Lu et al. [252] worked out the nonlinear plastic model
2 Size-Dependent Theories of Beams, Plates and Shells
and surface energy. Khajeansari et al. [244] investigated the bending deformation
of the Euler-Bernoulli beam lying on the Winkler-Pasternak foundation employing
theory of surface elasticity (see, for instance, [94, 245]). Simsek and Reddy [246]
proposed the functionally gradient model of a microbeam embedded into an elastic
medium using the modified couple stress theory (see [15, 247]) and the WinklerPasternak foundation. Limkatanyu et al. [248] proposed the Euler-Bernoulli beam
model supported by the Winkler-Pasternak foundation and included the microstructure and surface energy effects, which extended the non-classical model proposed
by Gao and Mahmuda [218]. More recently, Gao and Zhang [249] proposed the
non-classical model which takes into account the microstructure, surface energy
and influence of the foundation for the Kirchhoff plates with the use of variational
formulation.
Novel non-classical model of Kirchhoff plates with the use of the modified couple
stress theory, the surface theory of elasticity and two-parameter model of elastic foundation was presented in [249]. The variational formulation based on the Hamilton
principle which yielded the derived equations of motion and the boundary conditions was used. New plate model consisted of the scale material length parameter to
account of the microstructure effect, three elastic surface constants for description
of the surface energy, and two elastic moduli of foundation. In order to demonstrate
model abilities, the problem of static bending and free vibrations of the rectangular
plate simply supported was solved analytically. In the case of the static bending, the
numerical results indicate that in the considered cases deflection of the plate lying on
the elastic foundation is less than that forecasted by the classical models. Besides,
a difference in deflection forecast by new and classical models of plates is of large
amount if the plate thickness is efficiently small (it is decreased with increase of a
plate thickness). In the case of free vibrations, it was found that the eigenfrequencies
obtained for new plate models under elastic foundation or without are higher than
those obtained by the classical plate models. The so far described size effects in
microscale are validated by laboratory experiments. Besides, it was shown both analytically and numerically that an occurrence of elastic foundation implies decrease
of a plate deflection and increase of fundamental plate frequency, what was to be
expected. In the case of a rigid body with large ratio of the surface to its volume, the
size effects play a crucial role [37]. The size effects can be understood with the help
of theory of surface elasticity where locations of the atoms and material properties
are different in the surface and inside the body volume, and one surface effect is
included into artificially constructed member with a negligible thickness [87, 95,
96].
Theory of surface elasticity [61, 94] was employed for analysis of thin plates coupled with surface effects. For instance, Miller and Shenoy [37] developed a model
suitable for the description of the dependence of effective stiffness of nanosized
structural element (rod, beam or plate) on the size. Lim and He [250] proposed a
geometrically nonlinear model of nanoplates and nanomembranes based on Kirchhoff’s hypothesis and von Kármán deformations. Lu et al. [251] developed the model
of a thin plate dependent on the size with inclusion of normal stresses out and inside
of the surface of volume layer. Lu et al. [252] worked out the nonlinear plastic model