262
7 Mathematical Models of Functionally Graded Beams in Temperature Field
A general theory of static/dynamic behaviour of the three-layer structural construction with respect to their size dependent behaviour subjected to various external
loading has been proposed by Grigolyuk and Chulkov [76].
Remarkably, three-layer constructions do not only imply the structural nonhomogeneity regarding the beam thickness, but they require inclusions of the middle
layer due to occurrence of the transversal shear and compression, as well as coupling
effects of the layers should be taken into account. Grigolyuk and Chulkov [76] introduced a hypothesis of the linear distribution of the tangent displacements with respect
to both weight of the beam package and condition of the lack of compression in the
structural package while constructing their theory of three-layered structures. On
contrary to Bernoulli-Euler hypothesis, in Grigolyuk-Chulkov model a perpendicular line to the initial surface becomes not perpendicular to the deformed surface, since
it rotates on amount of a certain angle due to transversal shear generated by the middle layer. On the other hand, the external layers fit with Bernoulli-Euler hypotheses,
whereas the internal layer follows Timoshenko hypothesis. It should be emphasized
that the mentioned theory generalizes the classical beam theory. Grigolyuk-Chulkov,
using the hypothesis of straight cross sections for the internal layer, constructed the
equilibrium equations and studied the stability and vibrations of the carrying load
by the three-layer beam. Namely, the external layers have been made from materials
of infinite stiffness against shear and transversal compression, whereas the internal
layer had the infinite stiffness against the transversal compression. The static and
dynamic behaviour of multi-layer beams based on Bernoulli-Euler and Timoshenko
hypotheses have been employed for the whole structural package in Refs. [43, 44,
102, 117, 133, 171–174]. For multi-layer anisotropic plates and shells, Andreev and
Nemirovsky developed a general theory based on a broken line [175]. A regular and
chaotic contact/no-contact nonlinear dynamics of the multi-layer structure composed
of one plate and three Euler-Bernoulli beams coupled only by boundary condition
have been studied in the work of Awrejcewicz et al. [176].
The functional nanomaterials used as plies and layers put on the surfaces of
the rigid bodies essentially improve the exploitation characteristics of the industrial
products. If the multi-layer beams with the plane thin external layers do not have
the required loading ability, the latter can be increased/improved via application of
the reinforced external layering, i.e. employing the layers/plies/films having large
Young’s moduli in the form of nano-layers and the micro-layers made from carbon.
On the basis of the standard computations regarding the material resistance, the
thickness of the layers of a microbeam should achieve a tenth of microns in order
to satisfy the industrial requirements. However, as the experimental investigations
show, the mechanical properties of the micro- and nano-size elements depend on their
sizes. This is why a novel name has been introduced emphasizing the size-dependent
effect and characterizing the change of the properties of the structures composed
of elements of the size of microns and nanometres. Different features of the sizedependent effects exhibited by the micro- and nano-elements are widely described
in the existing literature, and among them, the gradient effects play a significant role.
The size-dependent behaviour of elastic elements have been observed experimentally while bending and turning of the microbeams [14, 16, 102, 173]. In Ref. [14],
7 Mathematical Models of Functionally Graded Beams in Temperature Field
A general theory of static/dynamic behaviour of the three-layer structural construction with respect to their size dependent behaviour subjected to various external
loading has been proposed by Grigolyuk and Chulkov [76].
Remarkably, three-layer constructions do not only imply the structural nonhomogeneity regarding the beam thickness, but they require inclusions of the middle
layer due to occurrence of the transversal shear and compression, as well as coupling
effects of the layers should be taken into account. Grigolyuk and Chulkov [76] introduced a hypothesis of the linear distribution of the tangent displacements with respect
to both weight of the beam package and condition of the lack of compression in the
structural package while constructing their theory of three-layered structures. On
contrary to Bernoulli-Euler hypothesis, in Grigolyuk-Chulkov model a perpendicular line to the initial surface becomes not perpendicular to the deformed surface, since
it rotates on amount of a certain angle due to transversal shear generated by the middle layer. On the other hand, the external layers fit with Bernoulli-Euler hypotheses,
whereas the internal layer follows Timoshenko hypothesis. It should be emphasized
that the mentioned theory generalizes the classical beam theory. Grigolyuk-Chulkov,
using the hypothesis of straight cross sections for the internal layer, constructed the
equilibrium equations and studied the stability and vibrations of the carrying load
by the three-layer beam. Namely, the external layers have been made from materials
of infinite stiffness against shear and transversal compression, whereas the internal
layer had the infinite stiffness against the transversal compression. The static and
dynamic behaviour of multi-layer beams based on Bernoulli-Euler and Timoshenko
hypotheses have been employed for the whole structural package in Refs. [43, 44,
102, 117, 133, 171–174]. For multi-layer anisotropic plates and shells, Andreev and
Nemirovsky developed a general theory based on a broken line [175]. A regular and
chaotic contact/no-contact nonlinear dynamics of the multi-layer structure composed
of one plate and three Euler-Bernoulli beams coupled only by boundary condition
have been studied in the work of Awrejcewicz et al. [176].
The functional nanomaterials used as plies and layers put on the surfaces of
the rigid bodies essentially improve the exploitation characteristics of the industrial
products. If the multi-layer beams with the plane thin external layers do not have
the required loading ability, the latter can be increased/improved via application of
the reinforced external layering, i.e. employing the layers/plies/films having large
Young’s moduli in the form of nano-layers and the micro-layers made from carbon.
On the basis of the standard computations regarding the material resistance, the
thickness of the layers of a microbeam should achieve a tenth of microns in order
to satisfy the industrial requirements. However, as the experimental investigations
show, the mechanical properties of the micro- and nano-size elements depend on their
sizes. This is why a novel name has been introduced emphasizing the size-dependent
effect and characterizing the change of the properties of the structures composed
of elements of the size of microns and nanometres. Different features of the sizedependent effects exhibited by the micro- and nano-elements are widely described
in the existing literature, and among them, the gradient effects play a significant role.
The size-dependent behaviour of elastic elements have been observed experimentally while bending and turning of the microbeams [14, 16, 102, 173]. In Ref. [14],