Ambiguous definitions of the piezoelectric coupling factor

When a piezoelectric material is subjected to a quasi-static electric (or mechanical) load, part of the input energy is converted to the mechanical (or electric) domain. The piezoelectric coupling factor k² is defined as the ratio between the converted energy and the supplied energy. This factor is often considered as a measure of the transduction efficiency of the material. Another definition of the coupling factor is a non-dimensional ratio of material coefficients. To examine the compatibility between these two different definitions, we consider several quasi-static loading cycles of a simple one-dimensional problem. We show that in some specific cases, the two definitions are equivalent, but that in other cases they are incompatible. In addition, we show that in specific quasi-static loading cycles, the converted energy may be increased by a slight modification of the unloading part of the cycle.