7.5 Stability of the Size-Dependent Graded Curvilinear Timoshenko Beams
231
7.5 Stability of the Size-Dependent and Functionally
Graded Curvilinear Timoshenko Beams
The size-dependent model is studied, based on the modified couple stress theory for
the geometrically nonlinear curvilinear Timoshenko beam made from a functionally
graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the
curvilinear beams is investigated with the use of the set-up (relaxation) method.
The second-order accuracy FDM is used to solve the problem of nonlinear PDEs
by reducing it to Cauchy problem. The obtained set of nonlinear ODEs is then
solved by the fourth-order Runge-Kutta method. The relaxation method is employed
to solve numerous static problems based on the dynamic approach. Eight different
combinations of size-dependent coefficients and the functionally graded material
coefficient are used to study the stress-strain responses of Timoshenko beams.
Stability loss of the curvilinear Timoshenko beams is investigated using Lyapunov
criterion based on the estimation of Lyapunov exponents. Beams with/without the
size-dependent behaviour, homogeneous beams and functionally graded beams having the same stiffness are investigated.
It is shown that in straight-line beams, the size-dependent effect decreases the
beam deflection. The same is observed if the most rigid layer is located on the top
of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid
layer essentially improves the beam strength against stability loss.
The observed transition of the largest Lyapunov exponent from a negative to
positive value corresponds to the transition from a pre-critical to post-critical beam
state.
7.5.1 Introduction
As already mentioned in Sect. 7.3.1, FGM can be composed by mixing two or more
materials with the functionally changed properties along a desirable direction [1].
They have a wide range of applications in both theoretical and industrial areas.
They can be used as materials for thermoisolation of cosmic structural elements in
the constructions of the nuclear reactors, and for fabrication of numerous sensors
and gyroscopes. In recent years, high interest has been observed in modelling and
investigating the functionally graded structures, in particular including their statics,
bending processes as well as dynamical characteristics [3–5].
It should be emphasized that in recent years the FGM have found direct applications in the micro- and nanostructures, like thin films/layers [7, 33], as well as in
the electromechanical/mechatronics systems [8, 9]. The size-dependent static and
dynamics behaviour of the microstructures has been approved experimentally [14,
16].
231
7.5 Stability of the Size-Dependent and Functionally
Graded Curvilinear Timoshenko Beams
The size-dependent model is studied, based on the modified couple stress theory for
the geometrically nonlinear curvilinear Timoshenko beam made from a functionally
graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the
curvilinear beams is investigated with the use of the set-up (relaxation) method.
The second-order accuracy FDM is used to solve the problem of nonlinear PDEs
by reducing it to Cauchy problem. The obtained set of nonlinear ODEs is then
solved by the fourth-order Runge-Kutta method. The relaxation method is employed
to solve numerous static problems based on the dynamic approach. Eight different
combinations of size-dependent coefficients and the functionally graded material
coefficient are used to study the stress-strain responses of Timoshenko beams.
Stability loss of the curvilinear Timoshenko beams is investigated using Lyapunov
criterion based on the estimation of Lyapunov exponents. Beams with/without the
size-dependent behaviour, homogeneous beams and functionally graded beams having the same stiffness are investigated.
It is shown that in straight-line beams, the size-dependent effect decreases the
beam deflection. The same is observed if the most rigid layer is located on the top
of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid
layer essentially improves the beam strength against stability loss.
The observed transition of the largest Lyapunov exponent from a negative to
positive value corresponds to the transition from a pre-critical to post-critical beam
state.
7.5.1 Introduction
As already mentioned in Sect. 7.3.1, FGM can be composed by mixing two or more
materials with the functionally changed properties along a desirable direction [1].
They have a wide range of applications in both theoretical and industrial areas.
They can be used as materials for thermoisolation of cosmic structural elements in
the constructions of the nuclear reactors, and for fabrication of numerous sensors
and gyroscopes. In recent years, high interest has been observed in modelling and
investigating the functionally graded structures, in particular including their statics,
bending processes as well as dynamical characteristics [3–5].
It should be emphasized that in recent years the FGM have found direct applications in the micro- and nanostructures, like thin films/layers [7, 33], as well as in
the electromechanical/mechatronics systems [8, 9]. The size-dependent static and
dynamics behaviour of the microstructures has been approved experimentally [14,
16].