we investigate the influence of scale length parameter, and non-homogeneity
coefficient on the dynamic characteristics and the scenario of transition from
periodic to chaotic beam vibrations. First, properties of various FGM materials are
revisited (Sect. 7.2) with an emphasis on the dependence of material properties on
temperature. Then, regular and chaotic vibrations of size-dependent Timoshenko
beams with functionally graded properties along their thickness are illustrated and
analysed (Sect. 7.3). The following new properties regarding the research topic
under consideration can be derived.
(i) We have considered the dynamics of nonlinear FG Timoshenko beams on
the basis of the modified couple stress theory, using a novel concept of the
bending line.
(ii) The influence of the size-dependent coefficient and the grading parameter on
the load-deflection dependence is investigated for the static problem. In order
to get solutions for nonlinear static problem, the relaxation method was
employed. It has been shown that the minimum deflection is achieved by the
beam when the functional grading process is taken into account and the
stiffer layer is located on the upper side in both cases, i.e. with/without the
size-dependent behaviour.
(iii) The functionally graded beam with the stiffer layer on the upper side is
suitable for application to carry dynamic loads for a given frequency and
amplitude of the harmonic excitation. This conclusion coincides with that
formulated for static problems. In the case of the homogeneous beam and the
beam with the stiffer layer located on the bottom side, the essential dependence of the obtained results on the size-dependent coefficient is observed.
(iv) In order to validate the reliability of the largest Lyapunov exponents (LLEs)
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-Takens-Newhouse 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
non-homogeneity. 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
xii
Introduction
coefficient on the dynamic characteristics and the scenario of transition from
periodic to chaotic beam vibrations. First, properties of various FGM materials are
revisited (Sect. 7.2) with an emphasis on the dependence of material properties on
temperature. Then, regular and chaotic vibrations of size-dependent Timoshenko
beams with functionally graded properties along their thickness are illustrated and
analysed (Sect. 7.3). The following new properties regarding the research topic
under consideration can be derived.
(i) We have considered the dynamics of nonlinear FG Timoshenko beams on
the basis of the modified couple stress theory, using a novel concept of the
bending line.
(ii) The influence of the size-dependent coefficient and the grading parameter on
the load-deflection dependence is investigated for the static problem. In order
to get solutions for nonlinear static problem, the relaxation method was
employed. It has been shown that the minimum deflection is achieved by the
beam when the functional grading process is taken into account and the
stiffer layer is located on the upper side in both cases, i.e. with/without the
size-dependent behaviour.
(iii) The functionally graded beam with the stiffer layer on the upper side is
suitable for application to carry dynamic loads for a given frequency and
amplitude of the harmonic excitation. This conclusion coincides with that
formulated for static problems. In the case of the homogeneous beam and the
beam with the stiffer layer located on the bottom side, the essential dependence of the obtained results on the size-dependent coefficient is observed.
(iv) In order to validate the reliability of the largest Lyapunov exponents (LLEs)
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-Takens-Newhouse 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
non-homogeneity. 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
xii
Introduction