7.4 Chaotic Dynamics of Size-Dependent Graded Timoshenko Beams
211
The classical theory of elasticity does not enable one to predict and account for
the size effect that occurs in the micro- and submicro-scale structures. To explain
these effects, various size-dependent continuum theories have been proposed, for
example, the couple stress elasticity [35, 36], nonlocal elasticity theory [103], strain
gradient elasticity [104] and surface elasticity [105].
In the couple stress theory, two additional material constants [35, 36] are distinguished. For isotropic elastic materials, they are related to the underlying microstructure of the material and are inherently difficult to determine. In Ref. [44], a modified
stress couple theory with only one additional parameter of the length scale of the
material has been developed. Such a model simplifies the use of a modified couple
stress theory.
In Refs. [43, 99, 106–109], the modified couple stress theory has been employed
to study static properties of Euler-Bernoulli beam. In Ref. [17], on the basis of the
modified couple stress theory and Hamilton’s principle, the equations of motion as
well as the boundary/initial conditions have been derived. In Ref. [38], the sizedependent behaviour of microbeams made of FGM has been investigated using the
modified couple stress theory for Euler-Bernoulli model.
Timoshenko beam model, contrarily Euler-Bernoulli model, allows one to take
into account the shear deformation and is more accurate for relatively thick beams and
beams of materials of low shear stiffness. The modified couple stress theory aimed at
investigating the size-dependent behaviour of the homogeneous Timoshenko beam
has been studied in Ref. [106]. In Ref. [63], these results have been extended for
Timoshenko beams made from FGM. Furthermore, in the case of statics, a closed
analytical formula has been obtained as well as the size-dependent frequencies have
been detected and studied.
The beams which are non-homogeneous with respect to their thickness have been
studied in Ref. [101] taking into account von Kármán nonlinearity and applying
the modified couple stress theory. The nonlinear PDEs governing the dynamics of
both Euler-Bernoulli and Timoshenko beams have been analysed. The investigations
regarding the influence of the internal material length scale parameter, non-constant
beam thickness properties, shear strain as well as the geometric nonlinearity on the
static beam deflections have been analysed.
It should be mentioned that the nonlinear PDEs governing the microstructural
dependences in beams made from FGM have been derived (up to the third-order
terms) in Ref. [64].
However, the critical literature overview implies that there is a lack of investigations of the nonlinear/chaotic dynamics of the beams made from FGM exhibiting, in
particular, the size-dependent behaviour. The above-mentioned stands for the motivation of the present section.
Problems devoted to solutions of nonlinear/chaotic continuous dynamical systems
[110–117] have been studied. In addition, the analysis of the nonlinear dynamics of
mechanical constructions has been conducted taking into account different types of
the nonlinearity, including geometrical, physical and fabrication ones, and including
actions of various physical fields like thermal, electric, magnetic and mechanical
[118, 119] as well as white-noise interaction [120].
211
The classical theory of elasticity does not enable one to predict and account for
the size effect that occurs in the micro- and submicro-scale structures. To explain
these effects, various size-dependent continuum theories have been proposed, for
example, the couple stress elasticity [35, 36], nonlocal elasticity theory [103], strain
gradient elasticity [104] and surface elasticity [105].
In the couple stress theory, two additional material constants [35, 36] are distinguished. For isotropic elastic materials, they are related to the underlying microstructure of the material and are inherently difficult to determine. In Ref. [44], a modified
stress couple theory with only one additional parameter of the length scale of the
material has been developed. Such a model simplifies the use of a modified couple
stress theory.
In Refs. [43, 99, 106–109], the modified couple stress theory has been employed
to study static properties of Euler-Bernoulli beam. In Ref. [17], on the basis of the
modified couple stress theory and Hamilton’s principle, the equations of motion as
well as the boundary/initial conditions have been derived. In Ref. [38], the sizedependent behaviour of microbeams made of FGM has been investigated using the
modified couple stress theory for Euler-Bernoulli model.
Timoshenko beam model, contrarily Euler-Bernoulli model, allows one to take
into account the shear deformation and is more accurate for relatively thick beams and
beams of materials of low shear stiffness. The modified couple stress theory aimed at
investigating the size-dependent behaviour of the homogeneous Timoshenko beam
has been studied in Ref. [106]. In Ref. [63], these results have been extended for
Timoshenko beams made from FGM. Furthermore, in the case of statics, a closed
analytical formula has been obtained as well as the size-dependent frequencies have
been detected and studied.
The beams which are non-homogeneous with respect to their thickness have been
studied in Ref. [101] taking into account von Kármán nonlinearity and applying
the modified couple stress theory. The nonlinear PDEs governing the dynamics of
both Euler-Bernoulli and Timoshenko beams have been analysed. The investigations
regarding the influence of the internal material length scale parameter, non-constant
beam thickness properties, shear strain as well as the geometric nonlinearity on the
static beam deflections have been analysed.
It should be mentioned that the nonlinear PDEs governing the microstructural
dependences in beams made from FGM have been derived (up to the third-order
terms) in Ref. [64].
However, the critical literature overview implies that there is a lack of investigations of the nonlinear/chaotic dynamics of the beams made from FGM exhibiting, in
particular, the size-dependent behaviour. The above-mentioned stands for the motivation of the present section.
Problems devoted to solutions of nonlinear/chaotic continuous dynamical systems
[110–117] have been studied. In addition, the analysis of the nonlinear dynamics of
mechanical constructions has been conducted taking into account different types of
the nonlinearity, including geometrical, physical and fabrication ones, and including
actions of various physical fields like thermal, electric, magnetic and mechanical
[118, 119] as well as white-noise interaction [120].