200
7 Mathematical Models of Functionally Graded Beams in Temperature Field
size-dependent phenomena was proposed by Eringen [22]. Based on the nonlocal
Eringen’s theory many investigations have been carried out aimed on work out of
nonlocal continuum models and employ them to study general behaviour of nanostructures [23–28] as well as problems related to bending effects of nanostructures
[29–32] and nanobeams [33, 34].
On the other hand, Toupin [35], Mindlin [36] and Koiter [37] developed the couple
stress theory successfully used while modelling of nanobeams. More recently, the
mentioned theory was employed by numerous researchers to study the size-dependent
dynamical behaviour of microstructures [17, 38–44]. Owing to the huge increase of
application of the micro-/nano-FGM structures [45–49] in recent years, the number of works devoted to analysis of the mechanical behaviour of such structures
has essentially increased [50–62]. The mentioned references present a comparison
of different values of the gradient material index between different beam models
constructed on a basis of the classical, modified couple stress and gradient theories.
Asghari et al. [38] investigated the size-dependent static and dynamic behaviour
of functionally graded microbeams based on the modified couple stress theory within
the elastic regime. Asghari et al. [63] analysed dynamic characteristics of the sizedependent FGM microbeams based on the strain gradient Timoshenko beam theory.
Asghari et al. [63] proposed the size-dependent governing equations of the FGM
Timoshenko beam based on the modified couple stress theory. Janghorban and Zare
[55] carried out free vibration study of functionally graded carbon nanotubes with
variable thickness taking into account Timoshenko beam theory. Kahrobaiyan et al.
[56] derived the size-dependent model of the FGM Euler-Bernoulli beam employing the theory of deformations gradient. The governing equations of motion were
obtained and the classical and non-classical boundary conditions were considered.
Both static and dynamic behaviour as well as free vibrations of the FGM beam simply
supported were studied. Simsek and Yurtcu [61] considered the static bending of the
FGM nanobeam based on the nonlocal Timoshenko and Euler-Bernoulli theories.
They succeeded in getting Navier type analytical solution for the simply supported
beam. Eltaher et al. [53] carried out an analysis of the free vibrations of FGM sizedependent nanobeams based on the nonlocal theory and Euler-Bernoulli beam theory.
Kiani [58] proposed a novel mathematical model for study vibrations and transversal
instabilities of moving nanoscale beam made from FGM. He derived equations of
motion of Rayleigh nanobeam by using nonlocal theory of elasticity. The frequencies of both longitudinal and transverse nanobeam vibrations were found by utilizing
Galerkin method and the associated modes of vibrations. Reddy [60] developed the
microstructure-dependent couple stress theories of functionally graded beams. He
used the principle of virtual displacements and he derived the size-dependent nonlinear PDEs for Euler-Bernoulli and Timoshenko beams with an account of von Kármán
nonlinearity. The influence of the scalar length parameter, the rule of the properties
of the beams change along with their thickness, shear deformation and geometric
nonlinearity and two static beam deflections were investigated.
Arbind et al. [64] utilized Hamilton principle to derive nonlinear PDEs for the
size-dependent stress-based third-order FGM beam theory. Ke et al. [65] investigated nonlinear free vibrations of FGM microbeams based on the modified couple
7 Mathematical Models of Functionally Graded Beams in Temperature Field
size-dependent phenomena was proposed by Eringen [22]. Based on the nonlocal
Eringen’s theory many investigations have been carried out aimed on work out of
nonlocal continuum models and employ them to study general behaviour of nanostructures [23–28] as well as problems related to bending effects of nanostructures
[29–32] and nanobeams [33, 34].
On the other hand, Toupin [35], Mindlin [36] and Koiter [37] developed the couple
stress theory successfully used while modelling of nanobeams. More recently, the
mentioned theory was employed by numerous researchers to study the size-dependent
dynamical behaviour of microstructures [17, 38–44]. Owing to the huge increase of
application of the micro-/nano-FGM structures [45–49] in recent years, the number of works devoted to analysis of the mechanical behaviour of such structures
has essentially increased [50–62]. The mentioned references present a comparison
of different values of the gradient material index between different beam models
constructed on a basis of the classical, modified couple stress and gradient theories.
Asghari et al. [38] investigated the size-dependent static and dynamic behaviour
of functionally graded microbeams based on the modified couple stress theory within
the elastic regime. Asghari et al. [63] analysed dynamic characteristics of the sizedependent FGM microbeams based on the strain gradient Timoshenko beam theory.
Asghari et al. [63] proposed the size-dependent governing equations of the FGM
Timoshenko beam based on the modified couple stress theory. Janghorban and Zare
[55] carried out free vibration study of functionally graded carbon nanotubes with
variable thickness taking into account Timoshenko beam theory. Kahrobaiyan et al.
[56] derived the size-dependent model of the FGM Euler-Bernoulli beam employing the theory of deformations gradient. The governing equations of motion were
obtained and the classical and non-classical boundary conditions were considered.
Both static and dynamic behaviour as well as free vibrations of the FGM beam simply
supported were studied. Simsek and Yurtcu [61] considered the static bending of the
FGM nanobeam based on the nonlocal Timoshenko and Euler-Bernoulli theories.
They succeeded in getting Navier type analytical solution for the simply supported
beam. Eltaher et al. [53] carried out an analysis of the free vibrations of FGM sizedependent nanobeams based on the nonlocal theory and Euler-Bernoulli beam theory.
Kiani [58] proposed a novel mathematical model for study vibrations and transversal
instabilities of moving nanoscale beam made from FGM. He derived equations of
motion of Rayleigh nanobeam by using nonlocal theory of elasticity. The frequencies of both longitudinal and transverse nanobeam vibrations were found by utilizing
Galerkin method and the associated modes of vibrations. Reddy [60] developed the
microstructure-dependent couple stress theories of functionally graded beams. He
used the principle of virtual displacements and he derived the size-dependent nonlinear PDEs for Euler-Bernoulli and Timoshenko beams with an account of von Kármán
nonlinearity. The influence of the scalar length parameter, the rule of the properties
of the beams change along with their thickness, shear deformation and geometric
nonlinearity and two static beam deflections were investigated.
Arbind et al. [64] utilized Hamilton principle to derive nonlinear PDEs for the
size-dependent stress-based third-order FGM beam theory. Ke et al. [65] investigated nonlinear free vibrations of FGM microbeams based on the modified couple