2.1 Plate/Shell Hypotheses and Applications to Linear Analysis
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thin structures, rather than thick structures. Using the classical displacement distribution assumption, the resulting strain-displacement relations contain second-order
derivative terms. Thus, it requires at least quadratic shape functions or higher-order
elements in FE analysis.
By replacing the three parameters in the classical plate theory with two parameters, bending and shear components, yields a two-variable refined plate theory (RPT),
which was developed by Shimpi and Patel [22]. The refined plate theory assumes
that the vertical displacement consists of bending and shear components. The displacement distribution is a third-order function of the position in the thickness direction. The transverse shear strains are no longer zero or constant, but a second-order
function of position in the thickness direction. Similar to the third-order shear deformation hypothesis, the transverse shear strains reach maximum at the mid-surface
and disappear at outer surfaces.
2.1.2 Reissner-Mindlin Hypothesis
Due to neglect of the transverse shear strains, the classical plate/shell theory is only
valid for thin structures. For moderately thick structures, transverse shear strains
should be included in the model. Accounting for transverse shear strains, ReissnerMindlin hypothesis was proposed and developed for plates and shells. The ReissnerMindlin hypothesis, known as first-order shear deformation (FOSD) hypothesis,
assumes that straight lines normal to the mid-surface remain straight after deformation, but not necessarily normal to the mid-surface. The FOSD hypothesis yields
constant transverse shear strains through the thickness. For more details of the FOSD
hypothesis for cylindrical and spherical shells, it refers to Ref. [23]. However, the
consideration of constant transverse shear strains are not always valid for plates and
shells, for example thick structures.
A large amount of publications have developed FE models based on the FOSD
hypothesis for smart structure. The first analytical FOSD model of piezoelectric
laminated plates was proposed and developed by Mindlin [24]. Later, the FOSD
hypothesis was implemented into piezoelectric integrated smart structures for static
analysis [25–28] and dynamic analysis [29–33]. Furthermore, a FOSD finite element
model was developed by Wang [34] for piezoelectric bimorph structures. A meshfree
model based on the FOSD hypothesis was developed by Liu et al. [35] for shape and
vibration control of laminated composite plates.
2.1.3 Higher-Order Shear Deformation Hypothesis
The Kirchhoff-Love hypothesis is valid for thin structures, while the FOSD hypothesis is applicable for moderately thick structures. This is because the zero or constant
transverse shear strains are not accurate enough for thick structures. The real sit-
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thin structures, rather than thick structures. Using the classical displacement distribution assumption, the resulting strain-displacement relations contain second-order
derivative terms. Thus, it requires at least quadratic shape functions or higher-order
elements in FE analysis.
By replacing the three parameters in the classical plate theory with two parameters, bending and shear components, yields a two-variable refined plate theory (RPT),
which was developed by Shimpi and Patel [22]. The refined plate theory assumes
that the vertical displacement consists of bending and shear components. The displacement distribution is a third-order function of the position in the thickness direction. The transverse shear strains are no longer zero or constant, but a second-order
function of position in the thickness direction. Similar to the third-order shear deformation hypothesis, the transverse shear strains reach maximum at the mid-surface
and disappear at outer surfaces.
2.1.2 Reissner-Mindlin Hypothesis
Due to neglect of the transverse shear strains, the classical plate/shell theory is only
valid for thin structures. For moderately thick structures, transverse shear strains
should be included in the model. Accounting for transverse shear strains, ReissnerMindlin hypothesis was proposed and developed for plates and shells. The ReissnerMindlin hypothesis, known as first-order shear deformation (FOSD) hypothesis,
assumes that straight lines normal to the mid-surface remain straight after deformation, but not necessarily normal to the mid-surface. The FOSD hypothesis yields
constant transverse shear strains through the thickness. For more details of the FOSD
hypothesis for cylindrical and spherical shells, it refers to Ref. [23]. However, the
consideration of constant transverse shear strains are not always valid for plates and
shells, for example thick structures.
A large amount of publications have developed FE models based on the FOSD
hypothesis for smart structure. The first analytical FOSD model of piezoelectric
laminated plates was proposed and developed by Mindlin [24]. Later, the FOSD
hypothesis was implemented into piezoelectric integrated smart structures for static
analysis [25–28] and dynamic analysis [29–33]. Furthermore, a FOSD finite element
model was developed by Wang [34] for piezoelectric bimorph structures. A meshfree
model based on the FOSD hypothesis was developed by Liu et al. [35] for shape and
vibration control of laminated composite plates.
2.1.3 Higher-Order Shear Deformation Hypothesis
The Kirchhoff-Love hypothesis is valid for thin structures, while the FOSD hypothesis is applicable for moderately thick structures. This is because the zero or constant
transverse shear strains are not accurate enough for thick structures. The real sit-