Three-dimensional exact solutions for the electromechanical response of triple-layer piezoelectric actuators

Actuators consisting of a metallic layer covered symmetrically by two transversely isotropic piezoelectric layers poled along the thickness direction are analyzed. By recasting the field equations of linear piezoelectricity into the transfer-matrix form and using a technique of expansion in a small parameter, the coupled electromechanical field in the actuator is expressed as a closed-form solution in terms of the three orthogonal displacement components on the mid-plane. The displacement components are governed by three two-dimensional differential equations, and the associated boundary conditions are specified in an average manner as in the classic plate theory. Solving these two-dimensional equations gives a three-dimensional solution for the piezoelectric actuator. Because the intermediate metallic layer results in some discontinuities in material properties and the electric displacement, significant physical considerations and analytical complexity arise while establishing the three-dimensional analytical method. As an example, the response of a cantilevered actuator subjected to an applied voltage is studied, and a significant result is discussed in detail. It is concluded that ignoring the electromechanical coupling leads to significant underestimation of the deformation of the actuator.