7.1 Introduction
199
stresses based on the introduced modified couple stress model are smaller compared
to the classical three-layer Grigolyuk-Chulkov beam model while increasing beam
thickness.
7.2 Literature Review
The functionally graded materials (FGM) can be fabricated as non-homogenous
composites, which are made from two or more materials with the required continuous
change of properties along a given direction [1]. The continuous change of the FGM
properties stands for their numerous engineering recognized advantages [2]. They
allow to omit problems associated with the occurrence of large shear stresses which
are likely to appear in multi-layer beams, stronger stability properties, the improved
heat resistance and decreased intensity of stresses. Nowadays, fabricated structures
made from FGM are widely applied in numerous branches of industry. The FGM
structures can be used as thermoisolation of the space engineering objects, nuclear
reactors, etc. This is why statics, bending and dynamic characteristics of structures
made from FGM have been intensively investigated [3–5]. In the recent years, FGM
found wide palette of applications in micro- and nanostructures such as thin films
[6, 7] and micro- and nano-electromechanical systems [8, 9].
The micro- and nanosize beams, plates and shells are widely used in the electromechanical systems (MEMS, NEMS) as the vibration sensors [10], micro-tubes
[11] and micro-switches [12]. The use of only one material does not allow to satisfy all material and economical requirements coming from the engineering world.
Witvrouw and Mehta [9] used FGM in order to satisfy the required mechanical, thermal and electric properties. They analysed the case when the material properties of
two-phase material made from poly-SiGe layers for MEMS/NEMS application of
thickness of microns and submicrons, and hence influence of the size effects cannot
be ignored. The size-dependent static and dynamic behaviour on the micro-level have
been validated experimentally [13–16]. The carried out experimental investigations
imply that the size-dependent behaviour is associated with internal materials property
when the characteristic size (thickness or diameter) is close to the internal material
length parameter [17]. Though numerous works devoted to that topic are based on
the linear models, which is contradicted to the experimental results pointing out the
necessity of inclusion of nonlinear features while investigating MEMS/NEMS [18].
Besides the existing experimental methods used for determination of mechanical
properties in the nanoscale, there are also a few theoretical approaches including the
molecular dynamics [19] and non-classical mechanics of continuum [20] for the purpose of modelling of micro- and nanostructures behaviour. Material properties in the
nanoscale depend on the object size, and consequently, the effect of the size dependence of the mechanical behaviour of nanomaterials should be taken into account
[21]. In order to overcome the mentioned problem, a few non-classical theories of
continuum have been employed which include the size-dependent effects while modelling nanostructures. One of the widely employed theories with an account of the
199
stresses based on the introduced modified couple stress model are smaller compared
to the classical three-layer Grigolyuk-Chulkov beam model while increasing beam
thickness.
7.2 Literature Review
The functionally graded materials (FGM) can be fabricated as non-homogenous
composites, which are made from two or more materials with the required continuous
change of properties along a given direction [1]. The continuous change of the FGM
properties stands for their numerous engineering recognized advantages [2]. They
allow to omit problems associated with the occurrence of large shear stresses which
are likely to appear in multi-layer beams, stronger stability properties, the improved
heat resistance and decreased intensity of stresses. Nowadays, fabricated structures
made from FGM are widely applied in numerous branches of industry. The FGM
structures can be used as thermoisolation of the space engineering objects, nuclear
reactors, etc. This is why statics, bending and dynamic characteristics of structures
made from FGM have been intensively investigated [3–5]. In the recent years, FGM
found wide palette of applications in micro- and nanostructures such as thin films
[6, 7] and micro- and nano-electromechanical systems [8, 9].
The micro- and nanosize beams, plates and shells are widely used in the electromechanical systems (MEMS, NEMS) as the vibration sensors [10], micro-tubes
[11] and micro-switches [12]. The use of only one material does not allow to satisfy all material and economical requirements coming from the engineering world.
Witvrouw and Mehta [9] used FGM in order to satisfy the required mechanical, thermal and electric properties. They analysed the case when the material properties of
two-phase material made from poly-SiGe layers for MEMS/NEMS application of
thickness of microns and submicrons, and hence influence of the size effects cannot
be ignored. The size-dependent static and dynamic behaviour on the micro-level have
been validated experimentally [13–16]. The carried out experimental investigations
imply that the size-dependent behaviour is associated with internal materials property
when the characteristic size (thickness or diameter) is close to the internal material
length parameter [17]. Though numerous works devoted to that topic are based on
the linear models, which is contradicted to the experimental results pointing out the
necessity of inclusion of nonlinear features while investigating MEMS/NEMS [18].
Besides the existing experimental methods used for determination of mechanical
properties in the nanoscale, there are also a few theoretical approaches including the
molecular dynamics [19] and non-classical mechanics of continuum [20] for the purpose of modelling of micro- and nanostructures behaviour. Material properties in the
nanoscale depend on the object size, and consequently, the effect of the size dependence of the mechanical behaviour of nanomaterials should be taken into account
[21]. In order to overcome the mentioned problem, a few non-classical theories of
continuum have been employed which include the size-dependent effects while modelling nanostructures. One of the widely employed theories with an account of the