Constitutive equations of symmetrical triple layer piezoelectric benders

Piezoelectric triple layer benders, with a structure of two piezoelectric top and bottom layers sandwiched by a non-piezoelectric elastic central layer, are one of the most commonly used piezoelectric devices. In this paper, we present the derivation of the constitutive equations of a symmetrical triple layer piezoelectric bender under different excitation conditions. The constitutive equations are presented by a 4x4 matrix with an external moment M, an external tip force F, a uniform load p, and an applied electric voltage V as the extensive parameters, with the generated tip angular deflection (slope) alpha, tip deflection delta, volume displacement v, and electric charge Q as the internal parameters. Further analysis on the electromechanical behavior of the triple layer piezoelectric bender can be made on the constitutive equations.