14
2 Literature Review
structures. Later, von Kármán type nonlinear FE models were developed based on
the classical plate theory [120], FOSD hypothesis [121] for buckling analysis of
piezoelectric structures. The von Kármán nonlinear FE models with FOSD hypothesis were studied by Panda and Ray [122], Varelis and Saravanos [123], for static
analysis of smart structures, and by Mukherjee and Chaudhuri [124] for dynamic
analysis.
Implementation of higher-order plate/shell hypothesis, Schmidt and Vu [125]
developed von Kármán nonlinear FE models based on both FOSD and TOSD
hypotheses for static and dynamic analysis of piezoelectric plates and shells. Cheng et
al. [126] carried out a similar study based on TOSD hypothesis for dynamic analysis. Shen and Yang [127], Singh et al. [128] developed nonlinear models using
higher-order through-thickness hypothesis.
Considering zigzag hypothesis for laminated structures, von Kármán nonlinear FE
models based on the first-order zigzag hypothesis were developed by Carrera [129],
Kapuria and Alam [130] for static analysis, and by Ray and Shivakumar [131],
Sarangi and Ray [132] for dynamic analysis. Furthermore, Icardi and Sciuva [133]
implemented a third-order zigzag hypothesis into geometrically nonlinear analysis
of piezoelectric structures.
2.3.2 Moderate Rotation Nonlinear Theory
The von Kármán type nonlinear theory is restricted to moderate rotations and small
displacements, since weak geometrical nonlinearity is included in the theory. Considering more nonlinear effects and under the assumption of moderate rotations, a
moderate rotation nonlinear shell theory was proposed and develop by Librescu and
Schmidt [62], Schmidt and Reddy [63] for composite lamination. The moderate rotation nonlinear theory considers more nonlinear effects, but the strain-displacement
terms are still limited. Therefore, the theory is classified into simplified nonlinear
theory. Afterwards, the theory was implemented into static and dynamic analysis of
smart structures by Lentzen and Schmidt [134], Lentzen et al. [135], Lentzen [80]
based on the FOSD hypothesis. Furthermore, for the purpose of comparison, moderate rotation theory with the FOSD hypothesis was investigated by Zhang and
Schmidt [136–138], Zhang [23] for static and dynamic analysis of smart structures.
2.3.3 Fully Geometrically Nonlinear Theory with Moderate
Rotations
Both von Kármán type nonlinear theory and moderate rotation nonlinear theory
consider limited nonlinear effects and under the assumption of small or moderate
rotations. To simulate structures with strong nonlinear effects, fully geometrically
2 Literature Review
structures. Later, von Kármán type nonlinear FE models were developed based on
the classical plate theory [120], FOSD hypothesis [121] for buckling analysis of
piezoelectric structures. The von Kármán nonlinear FE models with FOSD hypothesis were studied by Panda and Ray [122], Varelis and Saravanos [123], for static
analysis of smart structures, and by Mukherjee and Chaudhuri [124] for dynamic
analysis.
Implementation of higher-order plate/shell hypothesis, Schmidt and Vu [125]
developed von Kármán nonlinear FE models based on both FOSD and TOSD
hypotheses for static and dynamic analysis of piezoelectric plates and shells. Cheng et
al. [126] carried out a similar study based on TOSD hypothesis for dynamic analysis. Shen and Yang [127], Singh et al. [128] developed nonlinear models using
higher-order through-thickness hypothesis.
Considering zigzag hypothesis for laminated structures, von Kármán nonlinear FE
models based on the first-order zigzag hypothesis were developed by Carrera [129],
Kapuria and Alam [130] for static analysis, and by Ray and Shivakumar [131],
Sarangi and Ray [132] for dynamic analysis. Furthermore, Icardi and Sciuva [133]
implemented a third-order zigzag hypothesis into geometrically nonlinear analysis
of piezoelectric structures.
2.3.2 Moderate Rotation Nonlinear Theory
The von Kármán type nonlinear theory is restricted to moderate rotations and small
displacements, since weak geometrical nonlinearity is included in the theory. Considering more nonlinear effects and under the assumption of moderate rotations, a
moderate rotation nonlinear shell theory was proposed and develop by Librescu and
Schmidt [62], Schmidt and Reddy [63] for composite lamination. The moderate rotation nonlinear theory considers more nonlinear effects, but the strain-displacement
terms are still limited. Therefore, the theory is classified into simplified nonlinear
theory. Afterwards, the theory was implemented into static and dynamic analysis of
smart structures by Lentzen and Schmidt [134], Lentzen et al. [135], Lentzen [80]
based on the FOSD hypothesis. Furthermore, for the purpose of comparison, moderate rotation theory with the FOSD hypothesis was investigated by Zhang and
Schmidt [136–138], Zhang [23] for static and dynamic analysis of smart structures.
2.3.3 Fully Geometrically Nonlinear Theory with Moderate
Rotations
Both von Kármán type nonlinear theory and moderate rotation nonlinear theory
consider limited nonlinear effects and under the assumption of small or moderate
rotations. To simulate structures with strong nonlinear effects, fully geometrically