232
7 Mathematical Models of Functionally Graded Beams in Temperature Field
Unfortunately, no theory of the classical continuum can explain the size-dependent
behaviour exhibited on the micro- and submicrostructural scales. However, the nonclassical theories of continuum, like the strain gradient theory and the modified
couple stress theory, can be employed to study the size-dependent properties.
Timoshenko beam model [122], on contrary to Euler-Bernoulli model, takes into
account the shear deformation and can be modified to fit the real behaviour of the
constructions.
In Ref. [101], the functionally graded beams have been investigated taking into
account their thickness with and von Kármán-type nonlinearity. The size-dependent
behaviour has been studied with the help of the modified couple stress theory. Nonlinear PDEs governing the dynamics of Euler-Bernoulli and Timoshenko beams have
been derived and studied. An influence of the length parameter, the variation of the
beam properties along its thickness, the shear deformations as well as the influence
of the geometric nonlinearity on the beam static deflection have been investigated.
Free nonlinear vibrations of microbeams made of FGMs has been investigated in
Ref. [57] based on the modified couple stress theory accompanied by Timoshenko
beam theory and von Kármán geometric nonlinearity. It has been found that both
linear and nonlinear frequencies significantly increase when the thickness of the
FGM microbeam is comparable to the material length scale parameter. In Ref. [134]
dynamic stability of microbeams made of FGMs has been investigated based on the
modified couple stress theory and Timoshenko beam theory. The obtained results
show that the influence of the size-dependent effect on the dynamic stability characteristics is significant only when the thickness of the beam has a similar magnitude
as the material length scale parameter. The so far reported and overviewed studies
considered only straight beams and a displacement of the deflection curve has not
been taken into account that resulted in studying more complicated PDEs. On the
contrary, this problem has been addressed in this section and the obtained governing
PDEs have a simpler form comparing to those studied so far.
On the other hand, only a few investigations [50, 69, 70] present the analysis
of the curvilinear beams made from FGM in their transverse direction, taking into
account the size-dependent behaviour. In these works, the authors have studied only
the linear behaviour of FGM curved microbeams.
It should be emphasized that, according to the authors’ knowledge, there is a lack
of investigations of models of nonlinear curved microbeams made of functionally
graded materials in the existing literature.
The mentioned gaps in the research devoted to the size-dependent FGM structures
stand for motivation to analyse nonlinear vibrations of the curvilinear Timoshenkotype beams on a basis of the modified couple stress theory. The PDEs governing the
beam dynamics are derived and the influence of the size-dependent coefficient and
the material grading coefficient on the character of the load-deflection curve and the
plots of the beam forms are reported.
7 Mathematical Models of Functionally Graded Beams in Temperature Field
Unfortunately, no theory of the classical continuum can explain the size-dependent
behaviour exhibited on the micro- and submicrostructural scales. However, the nonclassical theories of continuum, like the strain gradient theory and the modified
couple stress theory, can be employed to study the size-dependent properties.
Timoshenko beam model [122], on contrary to Euler-Bernoulli model, takes into
account the shear deformation and can be modified to fit the real behaviour of the
constructions.
In Ref. [101], the functionally graded beams have been investigated taking into
account their thickness with and von Kármán-type nonlinearity. The size-dependent
behaviour has been studied with the help of the modified couple stress theory. Nonlinear PDEs governing the dynamics of Euler-Bernoulli and Timoshenko beams have
been derived and studied. An influence of the length parameter, the variation of the
beam properties along its thickness, the shear deformations as well as the influence
of the geometric nonlinearity on the beam static deflection have been investigated.
Free nonlinear vibrations of microbeams made of FGMs has been investigated in
Ref. [57] based on the modified couple stress theory accompanied by Timoshenko
beam theory and von Kármán geometric nonlinearity. It has been found that both
linear and nonlinear frequencies significantly increase when the thickness of the
FGM microbeam is comparable to the material length scale parameter. In Ref. [134]
dynamic stability of microbeams made of FGMs has been investigated based on the
modified couple stress theory and Timoshenko beam theory. The obtained results
show that the influence of the size-dependent effect on the dynamic stability characteristics is significant only when the thickness of the beam has a similar magnitude
as the material length scale parameter. The so far reported and overviewed studies
considered only straight beams and a displacement of the deflection curve has not
been taken into account that resulted in studying more complicated PDEs. On the
contrary, this problem has been addressed in this section and the obtained governing
PDEs have a simpler form comparing to those studied so far.
On the other hand, only a few investigations [50, 69, 70] present the analysis
of the curvilinear beams made from FGM in their transverse direction, taking into
account the size-dependent behaviour. In these works, the authors have studied only
the linear behaviour of FGM curved microbeams.
It should be emphasized that, according to the authors’ knowledge, there is a lack
of investigations of models of nonlinear curved microbeams made of functionally
graded materials in the existing literature.
The mentioned gaps in the research devoted to the size-dependent FGM structures
stand for motivation to analyse nonlinear vibrations of the curvilinear Timoshenkotype beams on a basis of the modified couple stress theory. The PDEs governing the
beam dynamics are derived and the influence of the size-dependent coefficient and
the material grading coefficient on the character of the load-deflection curve and the
plots of the beam forms are reported.