HSDT-layerwise analytical solution for rectangular piezoelectric laminated plates

The present paper develops a formulation for laminated plates with extensional distributed piezoelectric sensors/actuators. This formulation is based on linear electroelasticity, and an equivalent single layer is used for the mechanical displacement field, applying a Higher-Order Shear Deformation Theory (HSDT), whereas a layerwise discretization is used in the thickness direction for the electric potential. The electric and mechanical local equilibrium equations and local constitutive equations for the problem are identified. The Principle of Virtual Work is used to derive the dynamic equilibrium equations in terms of generalized forces and the consistent boundary conditions. The piezoelectric laminate constitutive equations are built and used to write the equations of motion in terms of generalized displacements. Finally, analytical solutions for simply supported square laminates with piezoelectric layers are developed. The entire laminate, composed of the base structure and piezoelectric layers, can be arbitrary orthotropic. The solution is adequate for an arbitrary number of piezoelectric layers and stacking positions. Moreover, the solution takes into account all material coefficients, whether mechanical, piezoelectric or dielectric. Analytical results are obtained for static bending, both in sensor and actuation modes, and for free vibration of symmetric cross-ply laminates with piezoelectric layers externally bonded to the plate.