198
7 Mathematical Models of Functionally Graded Beams in Temperature Field
(iv) In order to validate the reliability of the LLEs (largest Lyapunov exponents)
computed with Wolf’s algorithm, three qualitatively different methods have
been applied, i.e. Rosenstein’s, Kantz’s and the neural network one. The analysis implies that all methods give qualitatively the same result, i.e. positive or
negative values of the LLEs, over the whole time interval studied.
(v) For the considered values of the size-dependent and material grading parameters, the universal route to chaos, following the classical Ruelle-TakensNewhouse scenario, has been detected.
In Sect. 7.4, the size-dependent model based on a modified coupled stress theory has been constructed for the geometrically nonlinear curvilinear functionally
graded Timoshenko beams. To build the model, the concept of a reference line
was used. Compared to previous models for functionally graded beams, this model
is significantly simpler. In the case of the straight-line beams, the size-dependent
effect decreases the value of the beam deflection for the same coefficient of nonhomogeneity. It has been reported that the localization of the most rigid layer on the
upper side of the beam essentially decreases the deflections of the beam for the same
value of the size-dependent coefficients. In the case of the curvature, the most loading
ability is observed for the functionally graded beam with the most rigid layer located
on the beam top. In the case of the LE computations, in all 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 variants 2 and 8, there
are two positive LEs. It means that in these two cases, stiffer stability loss is exhibited. In Sect. 7.5, investigation of the 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. The last Sect. 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 simplest models and it includes the only one scalar length parameter.
However, it allows us to take into account the microstructural effects in both external
as well as 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
7 Mathematical Models of Functionally Graded Beams in Temperature Field
(iv) In order to validate the reliability of the LLEs (largest Lyapunov exponents)
computed with Wolf’s algorithm, three qualitatively different methods have
been applied, i.e. Rosenstein’s, Kantz’s and the neural network one. The analysis implies that all methods give qualitatively the same result, i.e. positive or
negative values of the LLEs, over the whole time interval studied.
(v) For the considered values of the size-dependent and material grading parameters, the universal route to chaos, following the classical Ruelle-TakensNewhouse scenario, has been detected.
In Sect. 7.4, the size-dependent model based on a modified coupled stress theory has been constructed for the geometrically nonlinear curvilinear functionally
graded Timoshenko beams. To build the model, the concept of a reference line
was used. Compared to previous models for functionally graded beams, this model
is significantly simpler. In the case of the straight-line beams, the size-dependent
effect decreases the value of the beam deflection for the same coefficient of nonhomogeneity. It has been reported that the localization of the most rigid layer on the
upper side of the beam essentially decreases the deflections of the beam for the same
value of the size-dependent coefficients. In the case of the curvature, the most loading
ability is observed for the functionally graded beam with the most rigid layer located
on the beam top. In the case of the LE computations, in all 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 variants 2 and 8, there
are two positive LEs. It means that in these two cases, stiffer stability loss is exhibited. In Sect. 7.5, investigation of the 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. The last Sect. 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 simplest models and it includes the only one scalar length parameter.
However, it allows us to take into account the microstructural effects in both external
as well as 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