The constituent equations of piezoelectric bimorphs

The constituent equations that describe the behavior of piezoelectric bimorphs for various mechanical boundary conditions are derived. The internal energy density of infinitesimally small volume elements in thermodynamic equilibrium is calculated in the presence of a voltage on the electrodes, a clamped cantilever condition on one side of the beam, and a set of three different classical boundary conditions on the other side of the beam: a mechanical moment M at the end of the beam, a force F perpendicular to the beam, applied at its tip, and a body force p. The total internal energy content is calculated by integrating over the entire volume of the beam. The canonical conjugate of the moment is calculated as the angular deflection at the tip of the beam, while that of the force at the tip is the local vertical deflection. The canonical conjugate of the uniform load on the beam proves to be the volume displacement V of the beam. The equations are given in the direct form, with internal parameters (M,V), (F,V), and (p ,V) as independent variables