|M.Sc Thesis||Department of Aerospace Engineering|
|Supervisor:||Assoc. Prof. Abramovich Haim|
Composite materials are commonly used as fundamental building blocks of smart structures with combinations of piezoelectric patches. One of the basic elements of adaptive structures is a thin composite plate with embedded patches that serve as actuators. Piezoelectric patches of extension type can be readily attached to the surface and under supplied voltage, actuate them by induced bending moments.
Analytical solutions for composite plates with arbitrary boundary conditions do not exist for piezoelectric or transverse loading conditions. Therefore, it becomes necessary to turn to one of the available approximate solutions, such as energy methods or the finite element method. However in order to understand the physical behavior of the response to applied loads, it is beneficial to obtain a closed form solution, which can serve as a reference for different numerical applications.
This work investigates flexural behavior of rectangular laminated orthotropic plates with attached piezoelectric layers and represents an iterative analytical method for obtaining highly accurate solutions for the plate deflection. The work is focused on thin piezolaminated orthotropic plate with combination of simply supported and clamped types of boundary conditions. In order to simplify the derivation of the equations, the present study is limited to symmetric and balanced laminates.
The equations were solved by the extended Kantorovich method (EKM), which is an iterative semi-analytical approach for solving partial differential equations. The iterative process converges rapidly. It was found that the final form of the generated solution is independent of the initial chosen functions. It is confirmed that these functions are neither required to satisfy the geometric nor the force boundary conditions, because the iterative procedure converges to a solution that satisfies all boundary conditions.
The derivation of the equations and the application of the Kantorovich routine was carried out by the symbolic commercial software MAPLE. At the beginning, the EKM method was validated by comparison to available analytical solutions. Isotropic and orthotropic plates of different thickness and dimensions were solved under transverse uniform loading. Then the case of piezoelectric loading was solved . The induced bending moment, which is usually applied by several piezoelectric patches, was assumed to be uniformly distributed over the plate surface. The solutions for thin piezolaminated orthotropic plates were compared to the results obtained using the commercial finite element code ANSYS, showing a good agreement.
The highly accurate results, obtained by the Kantorovich method, can be used as a benchmark for any numerical methods, for example the finite element analysis.