2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
57
cies. Zhang et al. [457] proposed a simple TSDT model for circular/annular FG
microplates earlier introduced by Thai and Kim [458], which included only four
unknowns. The DQ method was used to compute deflection, critical values and
eigenfrequencies of circular/annular FG microplates with various boundary conditions. Zhang et al. [459] worked out a simple HSDT model with deformation gradient
for the FG microplates based on the simple HSDT proposed by Thai and Choi [460–
463], which possesses only four unknowns (the authors considered an interaction
of the plate and elastic foundations). Akgoz and Civalek [464] developed the sinusoidal model of the deformation gradient for analysis of bending, stability and free
vibrations of the isotropic microplates based on the sinusoidal theory of Touratier
[371].
2.3.8.3 Surface Theory: RBT, TSDT, NSDT, HSDT
In work [255], an influence of the surface stresses on rigidity of the cantilever plates,
using full 3D model, was studied. It allowed to describe a link between the surface
stresses and stiffness of the cantilever, and then the cantilever behaviour versus the
size effects was studied.
In spite of the large account of the works published within the book research
topic, mainly linear models were used for computational analysis, though the
experimental results imply a need of account of the nonlinear features of the
micro/nanomechanical/thermomechanical systems [465].
This opens a novel challenging research field aimed on development of more precise nonlinear deformation models of the size-dependent beams subjected to static
and dynamic loads. To study dynamics of the size-dependent beams, there is a need
to use the apparatus of nonlinear dynamics including the Fourier and wavelet spectra,
phase portraits, Poincaré maps, the Lyapunov exponent’s computation (at least the
largest one), the autocorrelation functions, etc. The mentioned approach was earlier
employed by the authors of the given monograph to study nonlinear dynamics of
continuous systems in the form of beams, plates and shells [466–472]. There were
considered three types of nonlinearity, i.e. physical, geometric and design one (contact interaction in time). However, the mentioned results were obtained based on the
classical theory of elasticity, i.e. without considering the size-dependent behaviour
of constructions made from homogeneous materials.
The presented so far state of the art of literature shows that the investigations
of the size-dependent beams modelled via the Euler-Bernoulli, Timoshenko and
Sheremetev-Pelekh theories are based on the Duffing-type equations which are
yielded by the employment of Bubnov-Galerkin method in the first approximation
to the governing PDEs. There were considered linear problems aimed at estimation of eigenfrequencies and static problems focused on analysing the influence of
parameters responsible for the size effects.
In all cases, investigation of nonlinear static and dynamics, and in particular
chaotic dynamics, was not taken into account.
57
cies. Zhang et al. [457] proposed a simple TSDT model for circular/annular FG
microplates earlier introduced by Thai and Kim [458], which included only four
unknowns. The DQ method was used to compute deflection, critical values and
eigenfrequencies of circular/annular FG microplates with various boundary conditions. Zhang et al. [459] worked out a simple HSDT model with deformation gradient
for the FG microplates based on the simple HSDT proposed by Thai and Choi [460–
463], which possesses only four unknowns (the authors considered an interaction
of the plate and elastic foundations). Akgoz and Civalek [464] developed the sinusoidal model of the deformation gradient for analysis of bending, stability and free
vibrations of the isotropic microplates based on the sinusoidal theory of Touratier
[371].
2.3.8.3 Surface Theory: RBT, TSDT, NSDT, HSDT
In work [255], an influence of the surface stresses on rigidity of the cantilever plates,
using full 3D model, was studied. It allowed to describe a link between the surface
stresses and stiffness of the cantilever, and then the cantilever behaviour versus the
size effects was studied.
In spite of the large account of the works published within the book research
topic, mainly linear models were used for computational analysis, though the
experimental results imply a need of account of the nonlinear features of the
micro/nanomechanical/thermomechanical systems [465].
This opens a novel challenging research field aimed on development of more precise nonlinear deformation models of the size-dependent beams subjected to static
and dynamic loads. To study dynamics of the size-dependent beams, there is a need
to use the apparatus of nonlinear dynamics including the Fourier and wavelet spectra,
phase portraits, Poincaré maps, the Lyapunov exponent’s computation (at least the
largest one), the autocorrelation functions, etc. The mentioned approach was earlier
employed by the authors of the given monograph to study nonlinear dynamics of
continuous systems in the form of beams, plates and shells [466–472]. There were
considered three types of nonlinearity, i.e. physical, geometric and design one (contact interaction in time). However, the mentioned results were obtained based on the
classical theory of elasticity, i.e. without considering the size-dependent behaviour
of constructions made from homogeneous materials.
The presented so far state of the art of literature shows that the investigations
of the size-dependent beams modelled via the Euler-Bernoulli, Timoshenko and
Sheremetev-Pelekh theories are based on the Duffing-type equations which are
yielded by the employment of Bubnov-Galerkin method in the first approximation
to the governing PDEs. There were considered linear problems aimed at estimation of eigenfrequencies and static problems focused on analysing the influence of
parameters responsible for the size effects.
In all cases, investigation of nonlinear static and dynamics, and in particular
chaotic dynamics, was not taken into account.