Chapter 6
Nonlinear Analysis of Piezoceramic
Laminated Structures
Abstract In this chapter, geometrically nonlinear FE models, including RVK5,
MRT5, LRT5 and LRT56, are validated by static analysis of composite laminated
thin-walled structures. Later, the geometrically nonlinear FE models are test by buckling and post-buckling analysis of cylindrical composite panels. Afterwards, the geometrically nonlinear FE models are implemented into static and dynamic analysis
of piezolaminated beam, plate and shell structures. In the last part, the simulations
of nonlinear phenomena including both geometrically and electroelastic materially
nonlinear effects are performed through piezo structures undergoing large displacement and under strong electric field.
6.1 Benchmark Problems
6.1.1 Asymmetric Cross-Ply Laminated Plate
The first benchmark problem is an asymmetric cross-ply laminated plate, shown
in Fig. 6.1, proposed by Sun et al. [1], and studied later by Reddy [2], Basar et
al. [3], Kreja and Schmidt [4]. The composite plate consists of two substrate layers
with a stacking sequence of [90
◦ /0
◦ ]. The material angle 0
◦ represents the fiber reinforcement being along Θ
1 -axis, while material angle 90
◦ denotes that being along
Θ
2 -axis. The in-plane dimensions are 9 × 15 in
2 , and the thickness for each substrate layer is 0.02 in. The material properties are given in Table 6.1. The composite
structure is meshed by 9 × 2 quadrilateral shell elements along the Θ
1 - and Θ
2 -axis,
respectively. The boundary conditions of hinged and simply supported are considered. In the hinged case, all the DOFs are constrained except rotations around the
Θ
2 -axis. The simply supported boundary condition frees additionally the translational movement along the Θ
1 -direction based on the hinged case. A surface force is
applied on the plane with positive or negative, which is measured in the unit lb/in
2 .
The non-dimensional displacements (|w|/ h) of mid-point is calculated
For the hinged boundary condition, the mid-point displacements of the plate under
two loading cases ±q are derived using LRT56 theory, presented in Fig. 6.2a for small
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2021
S.-Q. Zhang, Nonlinear Analysis of Thin-Walled Smart Structures, Springer Tracts
in Mechanical Engineering, https://doi.org/10.1007/978-981-15-9857-9_6
101
Nonlinear Analysis of Piezoceramic
Laminated Structures
Abstract In this chapter, geometrically nonlinear FE models, including RVK5,
MRT5, LRT5 and LRT56, are validated by static analysis of composite laminated
thin-walled structures. Later, the geometrically nonlinear FE models are test by buckling and post-buckling analysis of cylindrical composite panels. Afterwards, the geometrically nonlinear FE models are implemented into static and dynamic analysis
of piezolaminated beam, plate and shell structures. In the last part, the simulations
of nonlinear phenomena including both geometrically and electroelastic materially
nonlinear effects are performed through piezo structures undergoing large displacement and under strong electric field.
6.1 Benchmark Problems
6.1.1 Asymmetric Cross-Ply Laminated Plate
The first benchmark problem is an asymmetric cross-ply laminated plate, shown
in Fig. 6.1, proposed by Sun et al. [1], and studied later by Reddy [2], Basar et
al. [3], Kreja and Schmidt [4]. The composite plate consists of two substrate layers
with a stacking sequence of [90
◦ /0
◦ ]. The material angle 0
◦ represents the fiber reinforcement being along Θ
1 -axis, while material angle 90
◦ denotes that being along
Θ
2 -axis. The in-plane dimensions are 9 × 15 in
2 , and the thickness for each substrate layer is 0.02 in. The material properties are given in Table 6.1. The composite
structure is meshed by 9 × 2 quadrilateral shell elements along the Θ
1 - and Θ
2 -axis,
respectively. The boundary conditions of hinged and simply supported are considered. In the hinged case, all the DOFs are constrained except rotations around the
Θ
2 -axis. The simply supported boundary condition frees additionally the translational movement along the Θ
1 -direction based on the hinged case. A surface force is
applied on the plane with positive or negative, which is measured in the unit lb/in
2 .
The non-dimensional displacements (|w|/ h) of mid-point is calculated
For the hinged boundary condition, the mid-point displacements of the plate under
two loading cases ±q are derived using LRT56 theory, presented in Fig. 6.2a for small
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2021
S.-Q. Zhang, Nonlinear Analysis of Thin-Walled Smart Structures, Springer Tracts
in Mechanical Engineering, https://doi.org/10.1007/978-981-15-9857-9_6
101