7.6 Stability of Curvilinear Euler-Bernoulli Beams in Temperature Fields
261
Figure 7.24 reports the influence of the temperature field intensity T (w) on the
beam stability loss for different geometric parameters λ = 50 and λ = 100.
In the case of the types 1–5 of the temperature field (see Table 7.14), for q = 0
and k x = 0, an increase in the temperature action implies a rapid change in the beam
deflections. An increase in the parameter λ yields an increase in the critical intensity
of the temperature input for which loss of stability occurs. For the type 4 of the
temperature field, one may conclude based on the reported results that the intensity
of the temperature action as well as the external load imply change in the beam form.
7.7 Mathematical Model of Three-Layer Micro- and
Nano-Beams Based on the Hypotheses of the
Grigolyuk-Chulkov and the Modified Couple Stress
Theory
The mathematical model of three-layered beams developed based on the hypothesis of the Grigolyuk-Chulkov and the modified couple stress theory and the sizedependent equations governing the layers motions on the micro- and nanoscales is
constructed. Hamilton’s principle yields the novel equations of motion as well as the
boundary/initial conditions regarding beams displacement. The latter ones clearly
exhibit the size dependent dynamics of the studied micro- and nanobeams, and the
introduced theory overlaps with the classical beam equations for large enough layer
thickness. In particular, a three-layer beam with the microlayer thickness has been
investigated with respect to the classical theory of Grigolyuk-Chulkov. The derived
boundary problem is of sixth order and can be solved analytically in the case of
statics. The carried out numerical experiments allowed to detect and explain sizedependent effects exhibited by the microbeams. The beam deflections and stress
yielded by the employed couple stress model are less than those predicted by the
classical Grigolyuk-Chulkov theory, while the estimated eigenfrequencies are higher,
respectively. It has been shown that the proposed model can be reduced to the classical three-layer Grigolyuk-Chulkov beam through increase of the layers thickness,
which validates our approach [170].
7.7.1 Introduction
It is well known that the three-layer structures have found applications in the mechanical and civil engineering, aviation, ship design, as well as in the airplanes and cosmic
industries. In recent years, they are employed in the design of numerous micro- and
nano-devices, including the micro- and nano-sensors as well as the electromagnetic
sensors.
261
Figure 7.24 reports the influence of the temperature field intensity T (w) on the
beam stability loss for different geometric parameters λ = 50 and λ = 100.
In the case of the types 1–5 of the temperature field (see Table 7.14), for q = 0
and k x = 0, an increase in the temperature action implies a rapid change in the beam
deflections. An increase in the parameter λ yields an increase in the critical intensity
of the temperature input for which loss of stability occurs. For the type 4 of the
temperature field, one may conclude based on the reported results that the intensity
of the temperature action as well as the external load imply change in the beam form.
7.7 Mathematical Model of Three-Layer Micro- and
Nano-Beams Based on the Hypotheses of the
Grigolyuk-Chulkov and the Modified Couple Stress
Theory
The mathematical model of three-layered beams developed based on the hypothesis of the Grigolyuk-Chulkov and the modified couple stress theory and the sizedependent equations governing the layers motions on the micro- and nanoscales is
constructed. Hamilton’s principle yields the novel equations of motion as well as the
boundary/initial conditions regarding beams displacement. The latter ones clearly
exhibit the size dependent dynamics of the studied micro- and nanobeams, and the
introduced theory overlaps with the classical beam equations for large enough layer
thickness. In particular, a three-layer beam with the microlayer thickness has been
investigated with respect to the classical theory of Grigolyuk-Chulkov. The derived
boundary problem is of sixth order and can be solved analytically in the case of
statics. The carried out numerical experiments allowed to detect and explain sizedependent effects exhibited by the microbeams. The beam deflections and stress
yielded by the employed couple stress model are less than those predicted by the
classical Grigolyuk-Chulkov theory, while the estimated eigenfrequencies are higher,
respectively. It has been shown that the proposed model can be reduced to the classical three-layer Grigolyuk-Chulkov beam through increase of the layers thickness,
which validates our approach [170].
7.7.1 Introduction
It is well known that the three-layer structures have found applications in the mechanical and civil engineering, aviation, ship design, as well as in the airplanes and cosmic
industries. In recent years, they are employed in the design of numerous micro- and
nano-devices, including the micro- and nano-sensors as well as the electromagnetic
sensors.