studied cases we have observed a transition from the negative to positive LE value,
which is associated with the transition from the pre-critical to post-critical beam
state. In the variants 2 and 8, there are two positive LE. It means that in these two
cases, stiffer stability loss is exhibited. In Sect. 7.5, the investigation of stability of
flexible curvilinear Euler-Bernoulli beams in a temperature field has been carried
out without any restrictions regarding the temperature field distribution. It has been
shown that the occurrence of imperfections due to either beam curvature or external
load implies a different form of beam stability loss while increasing the temperature
intensity. The type of temperature field has an essential impact on the beam stability
loss regarding the temperature intensity and external loading. Inclusion of the beam
curvature in the heat transfer equation yields an increase of the critical load
responsible for the stability loss as well as changes of the beam form regarding its
pre-critical state. Section 7.6 is devoted to the study of the mathematical model of a
three-layer micro- and nanobeams. Based on both Grigolyuk-Chulkov and modified
couple stress theories, the new model validated by both static and dynamic analyses
of the three-layer microbeams including only one scalar/length parameter has been
constructed, which takes into account the size effect. The employed Hamilton
principle yielded the governing equation of motion as well as general boundary and
initial conditions regarding displacements formulated for the microbeams. The
proposed model of the microbeam deformation is one of the most simple models,
and it includes the only one scalar length parameter. However, it allows us to take
into account the microstructural effects in both external and internal beam layers for
any boundary conditions. The finally formulated boundary value problem is of the
sixth order, and in the case of the static problem it is solved analytically. The
numerical results show that the studied beam model can explain the scale effect
exhibited by the microbeams. The obtained deflections and stresses based on the
introduced modified couple stress model are smaller compared to the classical
three-layer Grigolyuk-Chulkov beam model while increasing beam thickness.
Thermoelastic vibrations of the Timoshenko microbeams based on the modified
couple stress theory are studied in Chap. 8. In particular, the dependence of the
quality factor of nonlinear microbeam resonators under thermoelastic damping for
Timoshenko beams with regard to geometric nonlinearity is analysed. The constructed mathematical model is based on the modified couple stress theory which
implies prediction of size-dependent effects in microbeam resonators. The Hamilton
principle yields coupled nonlinear thermoelastic PDEs governing dynamics of the
Timoshenko microbeams for both plane stresses and plane deformations. Nonlinear
thermoelastic vibrations are investigated analytically and numerically, and quality
factors of the resonators versus geometric and material microbeam properties are
estimated. Results are presented for gold microbeams for different ambient temperatures and different beam thicknesses, and they are compared with results
yielded by the classical theory of elasticity in linear/nonlinear cases. The most
important conclusions of our study are summarized in the following three points.
Introduction
xiii
which is associated with the transition from the pre-critical to post-critical beam
state. In the variants 2 and 8, there are two positive LE. It means that in these two
cases, stiffer stability loss is exhibited. In Sect. 7.5, the investigation of stability of
flexible curvilinear Euler-Bernoulli beams in a temperature field has been carried
out without any restrictions regarding the temperature field distribution. It has been
shown that the occurrence of imperfections due to either beam curvature or external
load implies a different form of beam stability loss while increasing the temperature
intensity. The type of temperature field has an essential impact on the beam stability
loss regarding the temperature intensity and external loading. Inclusion of the beam
curvature in the heat transfer equation yields an increase of the critical load
responsible for the stability loss as well as changes of the beam form regarding its
pre-critical state. Section 7.6 is devoted to the study of the mathematical model of a
three-layer micro- and nanobeams. Based on both Grigolyuk-Chulkov and modified
couple stress theories, the new model validated by both static and dynamic analyses
of the three-layer microbeams including only one scalar/length parameter has been
constructed, which takes into account the size effect. The employed Hamilton
principle yielded the governing equation of motion as well as general boundary and
initial conditions regarding displacements formulated for the microbeams. The
proposed model of the microbeam deformation is one of the most simple models,
and it includes the only one scalar length parameter. However, it allows us to take
into account the microstructural effects in both external and internal beam layers for
any boundary conditions. The finally formulated boundary value problem is of the
sixth order, and in the case of the static problem it is solved analytically. The
numerical results show that the studied beam model can explain the scale effect
exhibited by the microbeams. The obtained deflections and stresses based on the
introduced modified couple stress model are smaller compared to the classical
three-layer Grigolyuk-Chulkov beam model while increasing beam thickness.
Thermoelastic vibrations of the Timoshenko microbeams based on the modified
couple stress theory are studied in Chap. 8. In particular, the dependence of the
quality factor of nonlinear microbeam resonators under thermoelastic damping for
Timoshenko beams with regard to geometric nonlinearity is analysed. The constructed mathematical model is based on the modified couple stress theory which
implies prediction of size-dependent effects in microbeam resonators. The Hamilton
principle yields coupled nonlinear thermoelastic PDEs governing dynamics of the
Timoshenko microbeams for both plane stresses and plane deformations. Nonlinear
thermoelastic vibrations are investigated analytically and numerically, and quality
factors of the resonators versus geometric and material microbeam properties are
estimated. Results are presented for gold microbeams for different ambient temperatures and different beam thicknesses, and they are compared with results
yielded by the classical theory of elasticity in linear/nonlinear cases. The most
important conclusions of our study are summarized in the following three points.
Introduction
xiii