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7 Mathematical Models of Functionally Graded Beams in Temperature Field
7.4 Chaotic Dynamics of Size-Dependent Timoshenko
Beams with Functionally Graded Properties Along
Their Thickness
In this section, we consider the size-dependent model based on the modified couple
stress theory of geometrically nonlinear Timoshenko beams made from FGM. We
assume that the beam properties are changed along with the beam thickness. The
influence of size-dependent and functionally graded coefficients on the vibration
characteristics, scenarios of transition from regular to chaotic vibrations as well as
a series of static problems with an emphasis put on the load-deflection behaviour
are studied. Our theoretical/numerical analysis is supported by methods of nonlinear
dynamics and the qualitative theory of differential equations supplemented by Fourier
and wavelet spectra, phase portraits and Lyapunov exponents spectra estimated by
different algorithms, including Wolf’s, Rosenstein’s, Kantz’s and neural networks.
The obtained results show that the beams modelled by the modified couple stress
theory are stiffer in comparison to those modelled by the classical continuum theory
(this is valid for any beam thickness distribution). We have also detected and numerically validated a general scenario governing transition into chaotic vibrations, which
follows the classical Ruelle-Takens-Newhouse scenario for the considered values of
the size-dependent and grading parameters.
7.4.1 Introduction
A functionally graded material can be made by mixing two or more materials with the
required continuous properties along the desired direction [1]. Continuous changes
in the material properties of the FGM yield a lot of engineering benefits since this
approach allows one to avoid an occurrence of large shear stresses, which are typically observed while fabricating multi-layer beams. The mentioned materials can be
directly applied in the cosmic industry, the nuclear reactors, etc. It should be emphasized that the static bending as well as the dynamic characteristics of the structural
members made from FGM have been intensively studied recently [3–5, 71, 97–101].
Furthermore, in recent years, FGM have been widely applied in micro- and nanostructures, including thin films/layers [6, 7], and micro- and nano-electromechanical
systems [8, 9]. The thickness of beams used in the above-mentioned structures is of
the order of microns and submicrons, thus the influence of scale effects on the beam
behaviour can be significant.
The experiments described in the works [13, 15, 102] show that size effects
play an important role in materials with microscale structure, such as a thin copper
wire, single crystal silver, nickel or steel epoxy polymeric beams. Namely, this phenomenon is exhibited when the characteristic dimension of the beam, i.e. either the
beam thickness or size, is close to the internal material length scale parameter [17].
7 Mathematical Models of Functionally Graded Beams in Temperature Field
7.4 Chaotic Dynamics of Size-Dependent Timoshenko
Beams with Functionally Graded Properties Along
Their Thickness
In this section, we consider the size-dependent model based on the modified couple
stress theory of geometrically nonlinear Timoshenko beams made from FGM. We
assume that the beam properties are changed along with the beam thickness. The
influence of size-dependent and functionally graded coefficients on the vibration
characteristics, scenarios of transition from regular to chaotic vibrations as well as
a series of static problems with an emphasis put on the load-deflection behaviour
are studied. Our theoretical/numerical analysis is supported by methods of nonlinear
dynamics and the qualitative theory of differential equations supplemented by Fourier
and wavelet spectra, phase portraits and Lyapunov exponents spectra estimated by
different algorithms, including Wolf’s, Rosenstein’s, Kantz’s and neural networks.
The obtained results show that the beams modelled by the modified couple stress
theory are stiffer in comparison to those modelled by the classical continuum theory
(this is valid for any beam thickness distribution). We have also detected and numerically validated a general scenario governing transition into chaotic vibrations, which
follows the classical Ruelle-Takens-Newhouse scenario for the considered values of
the size-dependent and grading parameters.
7.4.1 Introduction
A functionally graded material can be made by mixing two or more materials with the
required continuous properties along the desired direction [1]. Continuous changes
in the material properties of the FGM yield a lot of engineering benefits since this
approach allows one to avoid an occurrence of large shear stresses, which are typically observed while fabricating multi-layer beams. The mentioned materials can be
directly applied in the cosmic industry, the nuclear reactors, etc. It should be emphasized that the static bending as well as the dynamic characteristics of the structural
members made from FGM have been intensively studied recently [3–5, 71, 97–101].
Furthermore, in recent years, FGM have been widely applied in micro- and nanostructures, including thin films/layers [6, 7], and micro- and nano-electromechanical
systems [8, 9]. The thickness of beams used in the above-mentioned structures is of
the order of microns and submicrons, thus the influence of scale effects on the beam
behaviour can be significant.
The experiments described in the works [13, 15, 102] show that size effects
play an important role in materials with microscale structure, such as a thin copper
wire, single crystal silver, nickel or steel epoxy polymeric beams. Namely, this phenomenon is exhibited when the characteristic dimension of the beam, i.e. either the
beam thickness or size, is close to the internal material length scale parameter [17].