50
2 Size-Dependent Theories of Beams, Plates and Shells
the FG beams. Ansari et al. [339] generalized their earlier work [338] into study
of free vibrations of curvilinear FG microbeams. Ansari et al. [340] also developed
the TBT model based on the modified strain gradient theory for the case of thermal
bending of the FG microbeams with different boundary conditions. More recently,
Ansari et al. [341] extended the TBT model of the strain gradient theory to study
both linear and nonlinear viscoelastic beams. It should be emphasized that Gholami
et al. [342] worked out the TBT model with deformations gradient in order to study
nonlinear stability and dynamics of the micro-FG switches followed by the work
of Mindlin [17]. Small-scale effects and Casimir forces were considered. Xie et al.
[343] employed the collocation method based on the indirect radial basics functions
to solve the EBT and TBT models regarding bending, stability loss and computation
of the eigenfrequencies of the FG microbeams under various boundary conditions. It
should be emphasized that in the previous work devoted to FG microbeams the size
length parameters were chosen constant along the microbeams thickness. Therefore,
Tajalli et al. [344] improved the previous models based on the strain gradient theory
through account of the changeable length parameter along beam thickness. Practical
investigations of the static and bending and free vibrations validated correctness of
the approach [344].
Nonlinear TBT model based on the modified strain gradient theory was worked
out by Ansari et al. [345], in order to study the influence of size length-dependent
parameter and initial geometric interpretations on stability of the FG microbeams.
Approximate solutions to the problems of stability loss of the FG microbeam for
different boundary conditions were obtained with the help of the DQ method. Ansari
et al. [346] supplemented their previous work [345] by considering the influence of
thermal effects.
One of the earlier FSDT models with the gradient of deformations was proposed
by Sahmani and Ansari [347] and Ansari et al. [348] for a study of free vibrations and thermal stability loss of the FG microplates. Sahmani and Ansari [349]
considered only simply supported plates, whereas Ansari et al. [350] considered
microplates with various boundary conditions and with the use of the DQ method.
Ansari et al. [351] developed the model of nonlinear modified gradient FSDT theory
to investigate stability of the annular FG microplates under thermal load. Ansari et
al. [352] extended their previous work [350] to include the influence of temperature
on bending, stability and free vibrations of the FG microplates for various boundary
conditions. Shenas and Malekzadeh [351] also studied the temperature influence on
free vibrations of the FG microplates for various boundary conditions. However,
they employed Chebyshev-Ritz method instead of the DQ approach as it was done
in the work of Ansari et al. [350]. Ansari et al. [352] developed the FSDT model
for circular/annular FG microplates for different boundary conditions with the use
of the DQ method.
Gholami et al. [353] worked out the FSDT shell model based on the modified strain
gradient elasticity theory for the cylindrical FG microshells. Analytical solutions
were reported for the critical load of simply supported cylindrical FG microshells
under axial compression. Zhang et al. [354] also worked out the FSDT shell model
based on the modified strain gradient elasticity theory for cylindrical FG microshells,
2 Size-Dependent Theories of Beams, Plates and Shells
the FG beams. Ansari et al. [339] generalized their earlier work [338] into study
of free vibrations of curvilinear FG microbeams. Ansari et al. [340] also developed
the TBT model based on the modified strain gradient theory for the case of thermal
bending of the FG microbeams with different boundary conditions. More recently,
Ansari et al. [341] extended the TBT model of the strain gradient theory to study
both linear and nonlinear viscoelastic beams. It should be emphasized that Gholami
et al. [342] worked out the TBT model with deformations gradient in order to study
nonlinear stability and dynamics of the micro-FG switches followed by the work
of Mindlin [17]. Small-scale effects and Casimir forces were considered. Xie et al.
[343] employed the collocation method based on the indirect radial basics functions
to solve the EBT and TBT models regarding bending, stability loss and computation
of the eigenfrequencies of the FG microbeams under various boundary conditions. It
should be emphasized that in the previous work devoted to FG microbeams the size
length parameters were chosen constant along the microbeams thickness. Therefore,
Tajalli et al. [344] improved the previous models based on the strain gradient theory
through account of the changeable length parameter along beam thickness. Practical
investigations of the static and bending and free vibrations validated correctness of
the approach [344].
Nonlinear TBT model based on the modified strain gradient theory was worked
out by Ansari et al. [345], in order to study the influence of size length-dependent
parameter and initial geometric interpretations on stability of the FG microbeams.
Approximate solutions to the problems of stability loss of the FG microbeam for
different boundary conditions were obtained with the help of the DQ method. Ansari
et al. [346] supplemented their previous work [345] by considering the influence of
thermal effects.
One of the earlier FSDT models with the gradient of deformations was proposed
by Sahmani and Ansari [347] and Ansari et al. [348] for a study of free vibrations and thermal stability loss of the FG microplates. Sahmani and Ansari [349]
considered only simply supported plates, whereas Ansari et al. [350] considered
microplates with various boundary conditions and with the use of the DQ method.
Ansari et al. [351] developed the model of nonlinear modified gradient FSDT theory
to investigate stability of the annular FG microplates under thermal load. Ansari et
al. [352] extended their previous work [350] to include the influence of temperature
on bending, stability and free vibrations of the FG microplates for various boundary
conditions. Shenas and Malekzadeh [351] also studied the temperature influence on
free vibrations of the FG microplates for various boundary conditions. However,
they employed Chebyshev-Ritz method instead of the DQ approach as it was done
in the work of Ansari et al. [350]. Ansari et al. [352] developed the FSDT model
for circular/annular FG microplates for different boundary conditions with the use
of the DQ method.
Gholami et al. [353] worked out the FSDT shell model based on the modified strain
gradient elasticity theory for the cylindrical FG microshells. Analytical solutions
were reported for the critical load of simply supported cylindrical FG microshells
under axial compression. Zhang et al. [354] also worked out the FSDT shell model
based on the modified strain gradient elasticity theory for cylindrical FG microshells,