The constituent equations of piezoelectric bimorphs

A brief review of applications of piezoelectric bimorphs is presented. The constituent equations which 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 beam condition on one side of the beam and a set of three different classical boundary conditions on the other side of the beam. These are a mechanical moment M at the end of the beam, a force F perpendicular to the beam, applied at its tip, and a uniformly distributed body force p. The total internal energy content is calculated by integrating over the entire volume of the beam. Two different beam configurations are considered: parallel polarizations of the two adjoining elements of the beam with an internal electrode; and antiparallel orientation without an internal electrode. The canonical conjugate of the moment is calculated as the angular deflection at the tip of the beam α, while that of the force at the tip is the local vertical deflection δ. The canonical conjugate of the uniform load on the beam is found to be the volume displacement V of the beam. The canonical conjugate of the voltage across the electrodes is the charge on the electrodes. The equations are given in the direct form, with external parameters (M, V), (F, V), and (p, V) as independent variables and also in a linear combination with (M, F, p, V) as variables. These constituent equations can be used to calculate the behavior of the bimorph under any condition that can be described as a linear combination of forces at the tip, moments at the tip and uniform loads on the entire beam. This allows us to use the bimorph as a black box, without having to consider its internal movement or charges.