7.2 Literature Review
201
stress theory and with an account of von Kármán nonlinearity. Non-classical beam
model was developed in the frame of Timoshenko model with the inclusion of sizedependent length parameter.
Asgharifard and Yazdi [52] studied nonlinear free vibration of Euler-Bernoulli
FGM beams with geometric nonlinearity and surface effects and their influence on
the frequencies of vibrations. They derived nonlinear second-order ODE with square
and cubic nonlinearity by employing free modes of vibrations of the corresponding
linear system. Simsek [62] investigated nonlocal effects exhibited by the free longitudinal vibrations of functionally graded tapered nanorods. Two different boundary
conditions were considered: both rod ends were clamped and one was clamped and
one was free. Nateghi et al. [59] employed the modified couple stress theory to
study size-dependent buckling of functionally graded microbeams. They considered
different beam theories and different boundary conditions to determine the role of
shear deformation on beams buckling. The derived differential equation was solved
through the general method of differential quadratures. Ansari et al. [50] investigated size-dependent free vibration of FGM microbeam using the modified strain
gradient elasticity theory. The authors worked out a new size-dependent model consisting of three internal length parameters in order to take into account the effects
of small scale. Eltaher et al. [54] carried out the free vibration analysis of FGM
size-dependent nanobeams based on the nonlocal theory and Bernoulli-Euler model
by considering small deformation. Ansari et al. [51] employed the gradient elasticity theory to study small-scale effects influence of nonlinear vibrations of FGM
Timoshenko microbeams with an account of von Kármán geometric nonlinearity.
Anjomshoa [66] derived a model of elastic medium based on the nonlocal elasticity theory to study stability loss of orthotropic circle and elliptic nanoplates under
uniform compression in the plate plane. Ansari et al. [65] investigated axial buckling
of single-walled carbon FGM nanotubes with different boundary conditions based
on the Rayleigh-Ritz method and molecular dynamics.
Rahmani and Jandaghian [67] employed a nonlocal third-order shear deformation
theory and Rayleigh-Ritz method to analyse the stability of FGM nanobeams. The
proposed non-classical model includes the scale length parameter and takes into
account the size effect. The beam was functionally graded in the thickness direction,
and Poisson coefficient was assumed to be constant. The minimum of potential
energy yielded the beam governing equations and boundary conditions. Influence
of the size-dependent length parameter, gradient parameter and the ratio of length
over thickness in the characteristics of stability of the FGM nanobeam with simple
support ends and clamping ends was investigated.
Curved FGM beams in the transversal direction with an account of size-dependent
behaviour are rather rarely investigated [68–70]. Xiang and Yang [71] considered
free and forced vibration of three-layer laminate Timoshenko beam of variable thickness and under heat condition. However, the study did not include the size effect.
Dehrouyeh-Semnani et al. [72] derived the size-dependent three-layer sandwich
beam model based on symmetric-deviatoric couple stress theory and investigated
influence of the length of layers on the damping characteristics of microbeams vibrations. The similar problem was also analysed in [73–75]. In all so far mentioned
works, the same Bernoulli-Euler and Timoshenko hypotheses have been employed
201
stress theory and with an account of von Kármán nonlinearity. Non-classical beam
model was developed in the frame of Timoshenko model with the inclusion of sizedependent length parameter.
Asgharifard and Yazdi [52] studied nonlinear free vibration of Euler-Bernoulli
FGM beams with geometric nonlinearity and surface effects and their influence on
the frequencies of vibrations. They derived nonlinear second-order ODE with square
and cubic nonlinearity by employing free modes of vibrations of the corresponding
linear system. Simsek [62] investigated nonlocal effects exhibited by the free longitudinal vibrations of functionally graded tapered nanorods. Two different boundary
conditions were considered: both rod ends were clamped and one was clamped and
one was free. Nateghi et al. [59] employed the modified couple stress theory to
study size-dependent buckling of functionally graded microbeams. They considered
different beam theories and different boundary conditions to determine the role of
shear deformation on beams buckling. The derived differential equation was solved
through the general method of differential quadratures. Ansari et al. [50] investigated size-dependent free vibration of FGM microbeam using the modified strain
gradient elasticity theory. The authors worked out a new size-dependent model consisting of three internal length parameters in order to take into account the effects
of small scale. Eltaher et al. [54] carried out the free vibration analysis of FGM
size-dependent nanobeams based on the nonlocal theory and Bernoulli-Euler model
by considering small deformation. Ansari et al. [51] employed the gradient elasticity theory to study small-scale effects influence of nonlinear vibrations of FGM
Timoshenko microbeams with an account of von Kármán geometric nonlinearity.
Anjomshoa [66] derived a model of elastic medium based on the nonlocal elasticity theory to study stability loss of orthotropic circle and elliptic nanoplates under
uniform compression in the plate plane. Ansari et al. [65] investigated axial buckling
of single-walled carbon FGM nanotubes with different boundary conditions based
on the Rayleigh-Ritz method and molecular dynamics.
Rahmani and Jandaghian [67] employed a nonlocal third-order shear deformation
theory and Rayleigh-Ritz method to analyse the stability of FGM nanobeams. The
proposed non-classical model includes the scale length parameter and takes into
account the size effect. The beam was functionally graded in the thickness direction,
and Poisson coefficient was assumed to be constant. The minimum of potential
energy yielded the beam governing equations and boundary conditions. Influence
of the size-dependent length parameter, gradient parameter and the ratio of length
over thickness in the characteristics of stability of the FGM nanobeam with simple
support ends and clamping ends was investigated.
Curved FGM beams in the transversal direction with an account of size-dependent
behaviour are rather rarely investigated [68–70]. Xiang and Yang [71] considered
free and forced vibration of three-layer laminate Timoshenko beam of variable thickness and under heat condition. However, the study did not include the size effect.
Dehrouyeh-Semnani et al. [72] derived the size-dependent three-layer sandwich
beam model based on symmetric-deviatoric couple stress theory and investigated
influence of the length of layers on the damping characteristics of microbeams vibrations. The similar problem was also analysed in [73–75]. In all so far mentioned
works, the same Bernoulli-Euler and Timoshenko hypotheses have been employed