7.6 Stability of Curvilinear Euler-Bernoulli Beams in Temperature Fields
247
thermal post-buckling deformations [24]. It has been shown that all frequencies of
an unbuckled beam continuously decrease when the temperature rises.
Dinzart et al. [146] have applied cyclic bending to study the thermo-mechanical
response of beams made of a thermoplastic polymer. The chosen beam material
was dependent on temperature and frequency, while the inertia effects have been
neglected. Stability of the steady-state solutions and the conditions for thermal runaway (the so-called thermo-mechanical instability) have been investigated.
Abbasi et al. [147] have found the analytical solution for a beam made of a
functionally graded material, based on the first-order shear deformation theory with
lateral thermal shock loads. It has been assumed that the material properties across
the beam thickness are within the volume fraction of the constitutive materials and
the solution obeys the coupled thermoelastic theory.
Beams made of FGM are widely employed in mechanical, aerospace, civil and
nuclear engineering, and hence their adaptability to thermal loading and thermal
shocks plays a crucial role in both engineering and science. Although the FGM
beams were initially aimed at constructing thermal barriers in aerospace structures
and fusion reactors, where one deals with the extremely high temperature and large
thermal gradients, this research area includes many open problems still waiting to
be solved.
Nonlinear mechanical properties of the FGM beams subjected to in-plane thermal
loads have been analysed by Ma and Lee [148], whereas Gupta et al. [149] have
studied the post-buckling behaviour of slender columns by means of employing the
concept of coupled displacement field.
One of the research directions while studying vibrating beams in a temperature
field is based on inclusion of the temperature dependence of the elastic moduli of
materials (usually, the majority of metals exhibit a decrease in the elastic moduli when
the temperature increases). Peng He et al. [150] have used an improved Timoshenko
model to study beams of various tapered shapes under linear axial temperature distribution. The influence of the axial temperature difference on natural frequencies and
modal shapes of the beams have been analysed. Static stability of a sandwich beam
with both the viscoelastic core and supports subjected to an axial pulsating load and
1D temperature gradient has been investigated by Nayak et al. [151]. Hamilton’s
energy principle yielded the governing equations and boundary conditions, and then
a set of Hill’s equations have been derived. The authors have studied effects of shear
parameter, geometric parameters and thermal gradient on the non-dimensional static
buckling zones.
The classical Euler-Bernoulli and Timoshenko models have been recently
employed while investigating the FGM beams, carbon nanotubes or micromechanical resonators taking into account thermal effects. Benzair et al. [152] have employed
the nonlocal Timoshenko beam model and Euler beam model to study vibrations of
single-walled carbon nanotubes (CNTs) with emphasis put on the thermal effect. The
research has been aimed at the wave dispersion caused by the rotary inertia, the shear
deformation and the nonlocal elasticity characterizing the microstructure of CNTs.
Free vibrations of statically thermal post-buckled FGM beams with surfacebounded piezoelectric layers subject to both temperature rise and voltage have been
247
thermal post-buckling deformations [24]. It has been shown that all frequencies of
an unbuckled beam continuously decrease when the temperature rises.
Dinzart et al. [146] have applied cyclic bending to study the thermo-mechanical
response of beams made of a thermoplastic polymer. The chosen beam material
was dependent on temperature and frequency, while the inertia effects have been
neglected. Stability of the steady-state solutions and the conditions for thermal runaway (the so-called thermo-mechanical instability) have been investigated.
Abbasi et al. [147] have found the analytical solution for a beam made of a
functionally graded material, based on the first-order shear deformation theory with
lateral thermal shock loads. It has been assumed that the material properties across
the beam thickness are within the volume fraction of the constitutive materials and
the solution obeys the coupled thermoelastic theory.
Beams made of FGM are widely employed in mechanical, aerospace, civil and
nuclear engineering, and hence their adaptability to thermal loading and thermal
shocks plays a crucial role in both engineering and science. Although the FGM
beams were initially aimed at constructing thermal barriers in aerospace structures
and fusion reactors, where one deals with the extremely high temperature and large
thermal gradients, this research area includes many open problems still waiting to
be solved.
Nonlinear mechanical properties of the FGM beams subjected to in-plane thermal
loads have been analysed by Ma and Lee [148], whereas Gupta et al. [149] have
studied the post-buckling behaviour of slender columns by means of employing the
concept of coupled displacement field.
One of the research directions while studying vibrating beams in a temperature
field is based on inclusion of the temperature dependence of the elastic moduli of
materials (usually, the majority of metals exhibit a decrease in the elastic moduli when
the temperature increases). Peng He et al. [150] have used an improved Timoshenko
model to study beams of various tapered shapes under linear axial temperature distribution. The influence of the axial temperature difference on natural frequencies and
modal shapes of the beams have been analysed. Static stability of a sandwich beam
with both the viscoelastic core and supports subjected to an axial pulsating load and
1D temperature gradient has been investigated by Nayak et al. [151]. Hamilton’s
energy principle yielded the governing equations and boundary conditions, and then
a set of Hill’s equations have been derived. The authors have studied effects of shear
parameter, geometric parameters and thermal gradient on the non-dimensional static
buckling zones.
The classical Euler-Bernoulli and Timoshenko models have been recently
employed while investigating the FGM beams, carbon nanotubes or micromechanical resonators taking into account thermal effects. Benzair et al. [152] have employed
the nonlocal Timoshenko beam model and Euler beam model to study vibrations of
single-walled carbon nanotubes (CNTs) with emphasis put on the thermal effect. The
research has been aimed at the wave dispersion caused by the rotary inertia, the shear
deformation and the nonlocal elasticity characterizing the microstructure of CNTs.
Free vibrations of statically thermal post-buckled FGM beams with surfacebounded piezoelectric layers subject to both temperature rise and voltage have been