132
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
and Pelekh [1] proposed in 1964 development of the Timoshenko theory aimed at
account not only of rotation of beam middle curve but also its curving. The transversal
tangential stresses are equal to zero on the low and upper panel surfaces. The latter
model can be recognized as third-order model. In the series of publications [2, 3],
the latter theory was used to compute beams, plates and shells. After 27 years, the
problem was revisited by Levinson [4] and Reddy [5]. This is why in this book, and
beyond, we refer to the name as the Sheremetev–Pelekh (SP) model.
Micro- and nano-sized beams, plates and shells are widely employed in the
micro- and nano-electromechanical systems such as the vibration sensors [6], microconductors [7] and micro-switches [8].
The dependence of the elastic behaviour on the body size in the microscale has
been observed experimentally in metals [9, 10], alloys [11], polymers [12] and crystals [13]. In spite of the numerous works on the mentioned topic, in general, the carried
out numerical analysis is based on the linear models though the experimental results
point out a need of account of nonlinear behaviour of micro- and nano-mechanical
systems [14].
The classical mechanics of a rigid body does not allow for a proper interpretation
and forecast of the size-dependent behaviour exhibited in structures of micro- and
submicroscales due to lack of a parameter responsible for the scale effects. In the
recent years, a big effort has been observed focused on developing various theories
for modelling scale effects occurred in continuum including coupled stress theory
of elasticity [15, 16], nonlocal theory of elasticity [17], gradient theory of elasticity
[18] and surface elasticity [19].
We focus on the works devoted investigations of the problems of theory of elasticity based on the couple stress-based gradient theory. The fundamental theoretical
background of the couple stress theory was presented by Yang [20]. He introduced,
in spite of two classical material Lamé constants, an additional material constant of a
higher order. Fleck and Hutchinson [21] employed the modified couple stress theory
of elasticity in order to account and explanation of the size-dependent behaviour. In
the recent years, the latter theory has been used by numerous researchers for a proper
interpretation of the size-dependent dynamic behaviour of microstructures [22–26].
One of the important aspects of the use of the couple stress theory of elasticity
relies on its application to the static and dynamic problems of beams. The latter
structural members are widely used in fabrication of nano sensor, nano conductors and nano switchers. In order to construct mathematical models of beams, the
hypotheses of zero-order approximation (Bernoulli–Euler), first-order approximation (Timoshenko) and third-order approximation (Sheremetev–Pelekh) are used.
Each of the mentioned hypotheses can be viewed as an approximation of the beam
treated as 2D and 3D body. Recall that the first approximation being based on the
Bernoulli–Euler hypotheses does not take into account curvature of a normal to the
beam axis.
In references [27, 28] with the help of modified couple stress theory of elasticity, the governing linear equations, the initial and boundary conditions of the
size-dependent Euler–Bernoulli model are derived. Influence of the size-dependent
length parameter on the static deformation and magnitude of the natural frequen-
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
and Pelekh [1] proposed in 1964 development of the Timoshenko theory aimed at
account not only of rotation of beam middle curve but also its curving. The transversal
tangential stresses are equal to zero on the low and upper panel surfaces. The latter
model can be recognized as third-order model. In the series of publications [2, 3],
the latter theory was used to compute beams, plates and shells. After 27 years, the
problem was revisited by Levinson [4] and Reddy [5]. This is why in this book, and
beyond, we refer to the name as the Sheremetev–Pelekh (SP) model.
Micro- and nano-sized beams, plates and shells are widely employed in the
micro- and nano-electromechanical systems such as the vibration sensors [6], microconductors [7] and micro-switches [8].
The dependence of the elastic behaviour on the body size in the microscale has
been observed experimentally in metals [9, 10], alloys [11], polymers [12] and crystals [13]. In spite of the numerous works on the mentioned topic, in general, the carried
out numerical analysis is based on the linear models though the experimental results
point out a need of account of nonlinear behaviour of micro- and nano-mechanical
systems [14].
The classical mechanics of a rigid body does not allow for a proper interpretation
and forecast of the size-dependent behaviour exhibited in structures of micro- and
submicroscales due to lack of a parameter responsible for the scale effects. In the
recent years, a big effort has been observed focused on developing various theories
for modelling scale effects occurred in continuum including coupled stress theory
of elasticity [15, 16], nonlocal theory of elasticity [17], gradient theory of elasticity
[18] and surface elasticity [19].
We focus on the works devoted investigations of the problems of theory of elasticity based on the couple stress-based gradient theory. The fundamental theoretical
background of the couple stress theory was presented by Yang [20]. He introduced,
in spite of two classical material Lamé constants, an additional material constant of a
higher order. Fleck and Hutchinson [21] employed the modified couple stress theory
of elasticity in order to account and explanation of the size-dependent behaviour. In
the recent years, the latter theory has been used by numerous researchers for a proper
interpretation of the size-dependent dynamic behaviour of microstructures [22–26].
One of the important aspects of the use of the couple stress theory of elasticity
relies on its application to the static and dynamic problems of beams. The latter
structural members are widely used in fabrication of nano sensor, nano conductors and nano switchers. In order to construct mathematical models of beams, the
hypotheses of zero-order approximation (Bernoulli–Euler), first-order approximation (Timoshenko) and third-order approximation (Sheremetev–Pelekh) are used.
Each of the mentioned hypotheses can be viewed as an approximation of the beam
treated as 2D and 3D body. Recall that the first approximation being based on the
Bernoulli–Euler hypotheses does not take into account curvature of a normal to the
beam axis.
In references [27, 28] with the help of modified couple stress theory of elasticity, the governing linear equations, the initial and boundary conditions of the
size-dependent Euler–Bernoulli model are derived. Influence of the size-dependent
length parameter on the static deformation and magnitude of the natural frequen-