266
7 Mathematical Models of Functionally Graded Beams in Temperature Field
In all mentioned works, on contrary to Grigolyuk-Chulkov hypothesis, only one
hypothesis, i.e. either the Bernoulli-Euler or Timoshenko assumptions (for all layers)
have been employed. It does not allow to take into account a large difference between
the thickness of the layers (e.g. when the external layers are made from the thin
emulsion membranes).
We have employed the modified couple stress theory in order to take into account
the size-dependent effects in the three-layer beam with both stiffening and softening
external layers. This choice is motivated by an observation that in all works using
the mentioned theory, the stiffening effect is exhibited to explain the size-dependent
effects (see, for instance, the Refs. [14, 16, 44, 172]). Owing to the experimental results published in the work [195], where, in Table 1, the size dependency of
micro/nanostructures is reported for different materials, the majority of the analysed
materials show stiffening effects.
In this section, we study the beam having the external layer made from copper (Cu)
and the internal layer made from epoxide tar. Owing to the mentioned Table 1, those
materials exhibit stiffening effects. On the other hand, a comparison of the couple
stress theory and nonlocal Eringen’s theory has been carried out by Tsiatas and Yiotis
[196], where the effects of the beam softening based on nonlocal Eringen’s theory
and the beam stiffening due to the modified couple stress theory have been reported.
In Ref. [185], it is pointed out while investigating Timoshenko beams dynamics that
the use of nonlocal Eringen’s theory implies a decrease in the frequencies of free
beam vibrations, which stands in contrast to the results obtained using the modified
couple stress theory.
Relevance of the gradient and nonlocal (integral) elastic models to include the
size-dependent effects is widely illustrated and discussed, for instance, in Ref. [197].
The gradient models are considered as weak nonlocal models. Furthermore, in many
cases, the nonlocal models may yield a paradox, since the obtained solutions based
on the classical theory are the same as in the case of the nonlocal theory, i.e. there
is a lack of the size-dependent effect. The mentioned paradox does not appear while
using the modified couple stress theory.
Here, we propose the theory of three-layer beams based on the hypotheses of
Grigolyuk and Chulkov [76] as well as the modified couple stress theory and the
derived size-dependent equations of motion for the micro- and nano-order thickness
of the layers. Hamilton principle yields new equations of motion as well as the
boundary and initial conditions regarding displacements for the microbeams. The
obtained equations allow to explain the size-dependent behaviour of a microbeam
and they coincide with the classical equations if the layer thickness becomes large
enough. A numerical example of computation of the three-layer beam with microlevel thickness of the layers has been given, and a comparison of its behaviour versus
the classical Grigolyuk-Chulkov theory has been conducted.
7 Mathematical Models of Functionally Graded Beams in Temperature Field
In all mentioned works, on contrary to Grigolyuk-Chulkov hypothesis, only one
hypothesis, i.e. either the Bernoulli-Euler or Timoshenko assumptions (for all layers)
have been employed. It does not allow to take into account a large difference between
the thickness of the layers (e.g. when the external layers are made from the thin
emulsion membranes).
We have employed the modified couple stress theory in order to take into account
the size-dependent effects in the three-layer beam with both stiffening and softening
external layers. This choice is motivated by an observation that in all works using
the mentioned theory, the stiffening effect is exhibited to explain the size-dependent
effects (see, for instance, the Refs. [14, 16, 44, 172]). Owing to the experimental results published in the work [195], where, in Table 1, the size dependency of
micro/nanostructures is reported for different materials, the majority of the analysed
materials show stiffening effects.
In this section, we study the beam having the external layer made from copper (Cu)
and the internal layer made from epoxide tar. Owing to the mentioned Table 1, those
materials exhibit stiffening effects. On the other hand, a comparison of the couple
stress theory and nonlocal Eringen’s theory has been carried out by Tsiatas and Yiotis
[196], where the effects of the beam softening based on nonlocal Eringen’s theory
and the beam stiffening due to the modified couple stress theory have been reported.
In Ref. [185], it is pointed out while investigating Timoshenko beams dynamics that
the use of nonlocal Eringen’s theory implies a decrease in the frequencies of free
beam vibrations, which stands in contrast to the results obtained using the modified
couple stress theory.
Relevance of the gradient and nonlocal (integral) elastic models to include the
size-dependent effects is widely illustrated and discussed, for instance, in Ref. [197].
The gradient models are considered as weak nonlocal models. Furthermore, in many
cases, the nonlocal models may yield a paradox, since the obtained solutions based
on the classical theory are the same as in the case of the nonlocal theory, i.e. there
is a lack of the size-dependent effect. The mentioned paradox does not appear while
using the modified couple stress theory.
Here, we propose the theory of three-layer beams based on the hypotheses of
Grigolyuk and Chulkov [76] as well as the modified couple stress theory and the
derived size-dependent equations of motion for the micro- and nano-order thickness
of the layers. Hamilton principle yields new equations of motion as well as the
boundary and initial conditions regarding displacements for the microbeams. The
obtained equations allow to explain the size-dependent behaviour of a microbeam
and they coincide with the classical equations if the layer thickness becomes large
enough. A numerical example of computation of the three-layer beam with microlevel thickness of the layers has been given, and a comparison of its behaviour versus
the classical Grigolyuk-Chulkov theory has been conducted.