154
8 Conclusion and Future Work
in 3D case of piezoelectricity were first presented. Later, the electro-mechanically
coupled constitutive equations were develop from 3D case to 2D plate and shell
structures, in which arbitrary fibrous reinforcement orientation is considered. Based
on the constitutive relations of fibrous reinforced composite materials, the constitutive
models were extended to macro-fiber composite materials for two typical modes,
i.e. MFC-d31 and MFC-d33. In order to compute piezoelectric materials under strong
electric field, electroelastic coupled materially nonlinear constitutive equations were
developed by remaining second-order terms in Taylor’s expansion.
In Chap. 5, geometrically nonlinear finite element models were developed for
piezoelectric bonded smart structures. Resultant stresses and strains were introduced
to reduce volume integration to surface integration. The geometrically nonlinear
model LRT56 not only has fully nonlinear strain-displacement relations, but also
considers large rotation of shell director, in which large rotation is expressed by
Euler angles. Furthermore, an eight-node shell element with 5 mechanical DOFs for
each node and 1 electric DOF for each element were developed. Using the Hamilton’s principle and the principle of virtual work, the dynamic and static equilibrium
equations were obtained. For nonlinear case, the Total Lagrangian formulations were
implemented for both geometrically and electroelastic materially nonlinear models.
In the final sections, various numerical algorithms including Newmark method, central difference algorithm, Newton-Raphson method and Riks-Wempner method were
developed.
The last part of the main chapters presented linear and nonlinear simulations for
piezoelectric and macro-fiber composite bonded smart structures. Two major parts
were investigated in this report, namely nonlinear analysis for piezoelectric integrated smart structures and nonlinear analysis for macro-fiber composite laminated
structures. To validate the nonlinear FE models, geometrically nonlinear simulation
of composite laminated structures were studied, including buckling analysis. Then
the geometrically nonlinear FE models were applied to compute piezoelectric laminated plates and shells, as well as the electroelastic materially nonlinear models. The
second part of simulation presented geometrically nonlinear analysis of macro-fiber
composite integrated structures.
From the results presented in Chaps. 6 and 7, it can be concluded that simplified
nonlinear shell theories, RVK5, MRT5, LRT5, will fail to predict both static and
dynamic response for composite and piezoelectric laminated thin-walled structures
in the range of large rotations. This is because only simplified nonlinear straindisplacement relations are considered in the RVK5 and MRT5 theories, and in addition, no proper rotation updating is possible in all these simplified nonlinear shell
theories. In the case of smart structures undergoing large deflections and rotations,
large rotation theory (LRT56) has to be considered. For these structures under strong
driving electric field, electroelastic nonlinear phenomenon influence much on structural response, which should be considered in the simulations.
8 Conclusion and Future Work
in 3D case of piezoelectricity were first presented. Later, the electro-mechanically
coupled constitutive equations were develop from 3D case to 2D plate and shell
structures, in which arbitrary fibrous reinforcement orientation is considered. Based
on the constitutive relations of fibrous reinforced composite materials, the constitutive
models were extended to macro-fiber composite materials for two typical modes,
i.e. MFC-d31 and MFC-d33. In order to compute piezoelectric materials under strong
electric field, electroelastic coupled materially nonlinear constitutive equations were
developed by remaining second-order terms in Taylor’s expansion.
In Chap. 5, geometrically nonlinear finite element models were developed for
piezoelectric bonded smart structures. Resultant stresses and strains were introduced
to reduce volume integration to surface integration. The geometrically nonlinear
model LRT56 not only has fully nonlinear strain-displacement relations, but also
considers large rotation of shell director, in which large rotation is expressed by
Euler angles. Furthermore, an eight-node shell element with 5 mechanical DOFs for
each node and 1 electric DOF for each element were developed. Using the Hamilton’s principle and the principle of virtual work, the dynamic and static equilibrium
equations were obtained. For nonlinear case, the Total Lagrangian formulations were
implemented for both geometrically and electroelastic materially nonlinear models.
In the final sections, various numerical algorithms including Newmark method, central difference algorithm, Newton-Raphson method and Riks-Wempner method were
developed.
The last part of the main chapters presented linear and nonlinear simulations for
piezoelectric and macro-fiber composite bonded smart structures. Two major parts
were investigated in this report, namely nonlinear analysis for piezoelectric integrated smart structures and nonlinear analysis for macro-fiber composite laminated
structures. To validate the nonlinear FE models, geometrically nonlinear simulation
of composite laminated structures were studied, including buckling analysis. Then
the geometrically nonlinear FE models were applied to compute piezoelectric laminated plates and shells, as well as the electroelastic materially nonlinear models. The
second part of simulation presented geometrically nonlinear analysis of macro-fiber
composite integrated structures.
From the results presented in Chaps. 6 and 7, it can be concluded that simplified
nonlinear shell theories, RVK5, MRT5, LRT5, will fail to predict both static and
dynamic response for composite and piezoelectric laminated thin-walled structures
in the range of large rotations. This is because only simplified nonlinear straindisplacement relations are considered in the RVK5 and MRT5 theories, and in addition, no proper rotation updating is possible in all these simplified nonlinear shell
theories. In the case of smart structures undergoing large deflections and rotations,
large rotation theory (LRT56) has to be considered. For these structures under strong
driving electric field, electroelastic nonlinear phenomenon influence much on structural response, which should be considered in the simulations.