298
8 Thermoelastic Vibrations of Timoshenko Microbeams
the lack of eigen size of length, the classical mechanical theories cannot give proper
interpretation and prediction of the observed size-dependent phenomena.
More recently, theories of the continuum of a higher order for prediction of the
mentioned size-dependent relations were developed [60]. In the 1960s, some of
researchers proposed the couple stress theory of elasticity which can be classified
as the non-classical theory [61]. The theory made it possible to interpret the sizedependent effects using two higher order material constants in state equations. A
simplification introduced by Yang et al. [62] to the scale complex relations of the
couple stress theory of elasticity resulted in a modified couple stress theory which is
suitable because of effects using only one scale parameter of the material length.
Recently, many researchers investigated non-classical theorems of continuum in
order to formulate and study the size-dependent mechanical behaviour of beams
and plates.
Tsiatas [55] proposed a novel model of Kirchhoff plates based on a modified
couple stress theory (MCST) which can take into account the size-dependent effects.
Park et al. [63] proposed the Euler-Bernoulli beam model based on MCST, and
they investigated the influence of the length material parameter on static mechanical
microbeam properties.
Kong et al. [64, 65] solved the dynamical problem of Euler-Bernoulli beams
based on the MCST and the gradient theory of elasticity and they demonstrated that
eigenfrequencies of the microbeams obtained with the help of MCST are higher
than those predicted by the classical theory of Euler-Bernoulli beams. Using MCST,
Kahrobaiyan et al. [66] and Asghari et al. [67] studied analytically the static and
dynamic behaviour of functionally graded microbeams and material of atomic force
microscope microcantilevers.
Ma et al. [68] used Hamilton principle and relations of the nonlocal Eringer’s
theory to develop nonlocal theories of the Euler-Bernoulli, Timoshenko, Reddy and
Levinson beams.
Besides, Wang et al. [60] proposed a microscale Timoshenko beam model based
on the theory of gradient deformation. Using the Hamilton principle they derived the
governing equations, initial and boundary conditions. The model took into account
Poisson’s effects, including three length material parameters, and consequently, it
exhibited size effects.
In recent years, a series of works have been devoted to the study of MCST of beam
and plates that also functionally graded taking into account shear deformations [35,
69–74]. Reddy [70] and Reddy and Kim [71] derived equations of beams and plates
based on MCST.
However, in general, there are not many works reported in the existing literature
devoted to the study of size-dependent effect in the problems of coupled thermoelasticity of microbeams. Guo and Rogerson [75] investigated the influence of the size
on thermoelastic coupling in clamped elastic prismatic beams. Sun et al. [48] developed a theory for the coupled thermoelastic microbeam problem using the hyperbolic
model of heat transfer. They demonstrated how the size effect generated by the thermoelastic coupling disappears when thickness of the microbeam exceeds its critical
value depending on the material properties and boundary conditions. Rezazadeh et
8 Thermoelastic Vibrations of Timoshenko Microbeams
the lack of eigen size of length, the classical mechanical theories cannot give proper
interpretation and prediction of the observed size-dependent phenomena.
More recently, theories of the continuum of a higher order for prediction of the
mentioned size-dependent relations were developed [60]. In the 1960s, some of
researchers proposed the couple stress theory of elasticity which can be classified
as the non-classical theory [61]. The theory made it possible to interpret the sizedependent effects using two higher order material constants in state equations. A
simplification introduced by Yang et al. [62] to the scale complex relations of the
couple stress theory of elasticity resulted in a modified couple stress theory which is
suitable because of effects using only one scale parameter of the material length.
Recently, many researchers investigated non-classical theorems of continuum in
order to formulate and study the size-dependent mechanical behaviour of beams
and plates.
Tsiatas [55] proposed a novel model of Kirchhoff plates based on a modified
couple stress theory (MCST) which can take into account the size-dependent effects.
Park et al. [63] proposed the Euler-Bernoulli beam model based on MCST, and
they investigated the influence of the length material parameter on static mechanical
microbeam properties.
Kong et al. [64, 65] solved the dynamical problem of Euler-Bernoulli beams
based on the MCST and the gradient theory of elasticity and they demonstrated that
eigenfrequencies of the microbeams obtained with the help of MCST are higher
than those predicted by the classical theory of Euler-Bernoulli beams. Using MCST,
Kahrobaiyan et al. [66] and Asghari et al. [67] studied analytically the static and
dynamic behaviour of functionally graded microbeams and material of atomic force
microscope microcantilevers.
Ma et al. [68] used Hamilton principle and relations of the nonlocal Eringer’s
theory to develop nonlocal theories of the Euler-Bernoulli, Timoshenko, Reddy and
Levinson beams.
Besides, Wang et al. [60] proposed a microscale Timoshenko beam model based
on the theory of gradient deformation. Using the Hamilton principle they derived the
governing equations, initial and boundary conditions. The model took into account
Poisson’s effects, including three length material parameters, and consequently, it
exhibited size effects.
In recent years, a series of works have been devoted to the study of MCST of beam
and plates that also functionally graded taking into account shear deformations [35,
69–74]. Reddy [70] and Reddy and Kim [71] derived equations of beams and plates
based on MCST.
However, in general, there are not many works reported in the existing literature
devoted to the study of size-dependent effect in the problems of coupled thermoelasticity of microbeams. Guo and Rogerson [75] investigated the influence of the size
on thermoelastic coupling in clamped elastic prismatic beams. Sun et al. [48] developed a theory for the coupled thermoelastic microbeam problem using the hyperbolic
model of heat transfer. They demonstrated how the size effect generated by the thermoelastic coupling disappears when thickness of the microbeam exceeds its critical
value depending on the material properties and boundary conditions. Rezazadeh et