Introduction
Currently, in connection with the rapid development of technology and instrumentation, an important issue stands for the creation of devices as small as possible.
Here, we should recall the words of the great American Nobel laureate in Physics,
Richard Feynman, who was the first to predict the fabrication of devices at the
nanolevel. He presented a lecture given at the annual American Physical Society
meeting at Caltech on December 29, 1959, entitled “There’s Plenty of Room at the
Bottom: An Invitation to Enter a New Field of Physics”. The word “bottom” in this
phrase means a world of very small sizes, which occupies the area defined by
nanometers (1 nm = 10
−9 m). The mechanics of a deformable solid can describe the
elements of small-sized devices, but already here theories of higher order should be
taken into account. At this stage in the development of continuum mechanics, a
number of elasticity theories have emerged to overcome the problem. These include
the modified couple stress theory of elasticity, the surface theory of elasticity, the
nonlocal theory of elasticity, the gradient elasticity theory and their modifications.
The first chapter (Chap. 1) of this monograph gives an overview of studies on the
dynamics of nanoshells, nanoplates and nanobeams in natural/thermal/electric/
magnetic fields, obtained on the basis of some of the above-listed theories. The
review of existing papers and monographs shows that the problem of analysing the
statics and dynamics of MEMS/NEMS devices is only at the initial stage of
development. Namely, most of the research is limited to the analysis of
Duffing-type equations, which are obtained using the Bubnov-Galerkin variational
method in the first approximation, thus searching for solutions of strongly reduced
order models with one or two degrees of freedom. At present, when studying the
problems of statics and dynamics of plates and shells, kinematic models of the first
and second approximations are mainly employed. Moreover, the majority of the
approaches are based on a linear formulation and deal with a small number of
degrees of freedom. Often, in the problems under consideration, the solution is not
entirely reliable. Furthermore, since it is obtained by a limited number of methods,
their convergence is not shown or proved. Thus, the nonlinear dynamics of
nanostructures has not been sufficiently investigated, and no reliable scenarios
of the transition from periodic to chaotic vibrations have been revealed. The use of
ix
Currently, in connection with the rapid development of technology and instrumentation, an important issue stands for the creation of devices as small as possible.
Here, we should recall the words of the great American Nobel laureate in Physics,
Richard Feynman, who was the first to predict the fabrication of devices at the
nanolevel. He presented a lecture given at the annual American Physical Society
meeting at Caltech on December 29, 1959, entitled “There’s Plenty of Room at the
Bottom: An Invitation to Enter a New Field of Physics”. The word “bottom” in this
phrase means a world of very small sizes, which occupies the area defined by
nanometers (1 nm = 10
−9 m). The mechanics of a deformable solid can describe the
elements of small-sized devices, but already here theories of higher order should be
taken into account. At this stage in the development of continuum mechanics, a
number of elasticity theories have emerged to overcome the problem. These include
the modified couple stress theory of elasticity, the surface theory of elasticity, the
nonlocal theory of elasticity, the gradient elasticity theory and their modifications.
The first chapter (Chap. 1) of this monograph gives an overview of studies on the
dynamics of nanoshells, nanoplates and nanobeams in natural/thermal/electric/
magnetic fields, obtained on the basis of some of the above-listed theories. The
review of existing papers and monographs shows that the problem of analysing the
statics and dynamics of MEMS/NEMS devices is only at the initial stage of
development. Namely, most of the research is limited to the analysis of
Duffing-type equations, which are obtained using the Bubnov-Galerkin variational
method in the first approximation, thus searching for solutions of strongly reduced
order models with one or two degrees of freedom. At present, when studying the
problems of statics and dynamics of plates and shells, kinematic models of the first
and second approximations are mainly employed. Moreover, the majority of the
approaches are based on a linear formulation and deal with a small number of
degrees of freedom. Often, in the problems under consideration, the solution is not
entirely reliable. Furthermore, since it is obtained by a limited number of methods,
their convergence is not shown or proved. Thus, the nonlinear dynamics of
nanostructures has not been sufficiently investigated, and no reliable scenarios
of the transition from periodic to chaotic vibrations have been revealed. The use of
ix