264
7 Mathematical Models of Functionally Graded Beams in Temperature Field
axial principle, where both bending and axial deformations, as well as Poisson effect,
have been taken into account.
In Ref. [101], solving equations for the models of functionally graded BernoulliEuler, Timoshenko and Reddy-Levinson beams with respect to their thickness on the
basis of the modified couple stress theory and taking into account the size-dependent
state equations have been derived and analysed.
Asghari et al. [39] utilized the MCST to study nonlinear behaviour of Timoshenko
hinged-hinged beam including the mid-plane stretching. In the case of static bending,
the nonlinear size-dependent phenomena has been studied numerically, whereas a
solution to the free vibrations problem has been solved analytically. The latter work
has been extended by Ke and Wang [134] and Ke et al. [57] to study nonlinear beam
vibrations with an emphasis put to the stability estimation while including the axial
displacement in their study.
Reddy [60] extended theories related to nonlinear Euler-Bernoulli and Timoshenko beams taking into account through-thickness power-law vibration of a twoconstituent material and moderate rotation of transverse normal through the von
Kármán nonlinear strain. The proposed model, based on a modified couple stress
theory, power-law variation of the material and von Kármán geometric nonlinearity uses only one material length scale parameter and captures the size effect in a
functionally graded material.
Santos and Reddy [185] presented comparison among classical elasticity, nonlocal elasticity and modified couple stress theories for free vibration analysis of
the Timoshenko beams, where the rotary inertia and nonlocal parameter have been
taken into account. The convergence of the theories has been demonstrated taking
into account an increase of the beam global dimension.
Reddy and Arbind [186] reformulated the classical beam theories, i.e. BernoulliEuler and Timoshenko, using a modified couple stress theory and employing
thickness power-law variation of a functionally graded material. The algebraic relationships have been derived for the beam deflections, slopes and stress resultants.
They have been validated through examples of straight beams with simply supported
and clamped boundary conditions.
The nonlinear resonant dynamics of a microscale beam based on the modified
couple stress theory has been analysed numerically in Ref. [137]. First, Hamilton’s
principle has been employed to derive a PDE governing motion using the modified
couple stress theory, and then Galerkin technique has been utilized to obtain a set of
coupled nonlinear ODEs. The effect of different system parameters on the resonant
dynamics system response has been studied.
Kahrobaiyan et al. [187] proposed a new comprehensive Timoshenko beam element based on the modified couple stress theory. Then the mass and stiffness matrices
have been computed using the energy approach and Hamilton’s principle. In particular, the static deflection of a short microbeam and pull-in voltage of an electrostatically
actuated microcantilever made of silicon are estimated using this new beam element.
A microstructure-dependent nonlinear third-order beam theory which accounts
for through-thickness power-law variation of a two-constituent material has been
developed by Arbind et al. [64] based on modified couple stress theory the influence
of the material length has been investigated.
7 Mathematical Models of Functionally Graded Beams in Temperature Field
axial principle, where both bending and axial deformations, as well as Poisson effect,
have been taken into account.
In Ref. [101], solving equations for the models of functionally graded BernoulliEuler, Timoshenko and Reddy-Levinson beams with respect to their thickness on the
basis of the modified couple stress theory and taking into account the size-dependent
state equations have been derived and analysed.
Asghari et al. [39] utilized the MCST to study nonlinear behaviour of Timoshenko
hinged-hinged beam including the mid-plane stretching. In the case of static bending,
the nonlinear size-dependent phenomena has been studied numerically, whereas a
solution to the free vibrations problem has been solved analytically. The latter work
has been extended by Ke and Wang [134] and Ke et al. [57] to study nonlinear beam
vibrations with an emphasis put to the stability estimation while including the axial
displacement in their study.
Reddy [60] extended theories related to nonlinear Euler-Bernoulli and Timoshenko beams taking into account through-thickness power-law vibration of a twoconstituent material and moderate rotation of transverse normal through the von
Kármán nonlinear strain. The proposed model, based on a modified couple stress
theory, power-law variation of the material and von Kármán geometric nonlinearity uses only one material length scale parameter and captures the size effect in a
functionally graded material.
Santos and Reddy [185] presented comparison among classical elasticity, nonlocal elasticity and modified couple stress theories for free vibration analysis of
the Timoshenko beams, where the rotary inertia and nonlocal parameter have been
taken into account. The convergence of the theories has been demonstrated taking
into account an increase of the beam global dimension.
Reddy and Arbind [186] reformulated the classical beam theories, i.e. BernoulliEuler and Timoshenko, using a modified couple stress theory and employing
thickness power-law variation of a functionally graded material. The algebraic relationships have been derived for the beam deflections, slopes and stress resultants.
They have been validated through examples of straight beams with simply supported
and clamped boundary conditions.
The nonlinear resonant dynamics of a microscale beam based on the modified
couple stress theory has been analysed numerically in Ref. [137]. First, Hamilton’s
principle has been employed to derive a PDE governing motion using the modified
couple stress theory, and then Galerkin technique has been utilized to obtain a set of
coupled nonlinear ODEs. The effect of different system parameters on the resonant
dynamics system response has been studied.
Kahrobaiyan et al. [187] proposed a new comprehensive Timoshenko beam element based on the modified couple stress theory. Then the mass and stiffness matrices
have been computed using the energy approach and Hamilton’s principle. In particular, the static deflection of a short microbeam and pull-in voltage of an electrostatically
actuated microcantilever made of silicon are estimated using this new beam element.
A microstructure-dependent nonlinear third-order beam theory which accounts
for through-thickness power-law variation of a two-constituent material has been
developed by Arbind et al. [64] based on modified couple stress theory the influence
of the material length has been investigated.