202
7 Mathematical Models of Functionally Graded Beams in Temperature Field
to the whole layers package, that did not allow to take into account a big difference
of the layers’ thickness (for instance, when upper and lower layers are made from
thin films).
Three-layer constructions have been employed in engineering for a long time in
aircraft/aviation engineering, ships design and civil engineering, machinery industries and space engineering. However, in recent years they are more often applied in
the construction of various micro- and nano-devices, for instance, micro- and nanobiosensors and micro- and nano-electromechanical sensors. The general theory of
material strength resistance of three-layer constructions of different types and subjected to external load have been worked out by Grigoluk and Chulkov [76]. Threelayer constructions require not only functional gradation along with their thickness
but also require taking into account the filling layer work and transversal shear and
transversal compression which implies to consider the coupling of layers. Grigolyuk
and Chulkov introduced a hypothesis of linear distribution of tangent displacements
along with the package height and lack of compression of the whole package what
served for the obtaining three-layer thin-walled construction. On the contrary to
Euler-Bernoulli hypothesis, a normal to the input surface undergoes rotation on the
amount of a certain angle due to action of the transversal shear of the filler. In other
words, external carrying load layers obey Bernoulli-Euler hypotheses, whereas the
internal layer (filler) satisfies Timoshenko hypotheses. Grigoluk and Chulkov, by
taking into account the straight cross section for the filler developed a general theory
for beams. Namely, they derived the governing beam behaviour PDEs and considered stability and vibration of three-layer beams with carrying load by different
layers made from materials of infinite stiffness with regard to shear and transversal
compression, whereas the filler satisfied condition of the infinite stiffness against the
transversal compression. Both static and dynamic behaviours of multi-layer beams
on the basis of Euler-Bernoulli and Timoshenko hypotheses were analysed in [77,
78].
7.3 Laws of Properties Change of FGM
In this section, we consider the most known laws of changes of properties of FGM
along either thickness or length of beam.
7.3.1 Properties of Material P-FGM
It is assumed that the volume amount of the FGM is graded along plate thickness
and obeys the following function:
ρ (z) =
z +
h / 2
h
p
,
(7.1)
7 Mathematical Models of Functionally Graded Beams in Temperature Field
to the whole layers package, that did not allow to take into account a big difference
of the layers’ thickness (for instance, when upper and lower layers are made from
thin films).
Three-layer constructions have been employed in engineering for a long time in
aircraft/aviation engineering, ships design and civil engineering, machinery industries and space engineering. However, in recent years they are more often applied in
the construction of various micro- and nano-devices, for instance, micro- and nanobiosensors and micro- and nano-electromechanical sensors. The general theory of
material strength resistance of three-layer constructions of different types and subjected to external load have been worked out by Grigoluk and Chulkov [76]. Threelayer constructions require not only functional gradation along with their thickness
but also require taking into account the filling layer work and transversal shear and
transversal compression which implies to consider the coupling of layers. Grigolyuk
and Chulkov introduced a hypothesis of linear distribution of tangent displacements
along with the package height and lack of compression of the whole package what
served for the obtaining three-layer thin-walled construction. On the contrary to
Euler-Bernoulli hypothesis, a normal to the input surface undergoes rotation on the
amount of a certain angle due to action of the transversal shear of the filler. In other
words, external carrying load layers obey Bernoulli-Euler hypotheses, whereas the
internal layer (filler) satisfies Timoshenko hypotheses. Grigoluk and Chulkov, by
taking into account the straight cross section for the filler developed a general theory
for beams. Namely, they derived the governing beam behaviour PDEs and considered stability and vibration of three-layer beams with carrying load by different
layers made from materials of infinite stiffness with regard to shear and transversal
compression, whereas the filler satisfied condition of the infinite stiffness against the
transversal compression. Both static and dynamic behaviours of multi-layer beams
on the basis of Euler-Bernoulli and Timoshenko hypotheses were analysed in [77,
78].
7.3 Laws of Properties Change of FGM
In this section, we consider the most known laws of changes of properties of FGM
along either thickness or length of beam.
7.3.1 Properties of Material P-FGM
It is assumed that the volume amount of the FGM is graded along plate thickness
and obeys the following function:
ρ (z) =
z +
h / 2
h
p
,
(7.1)