The computation of the k factor for lossy materials around resonance

An important property of a piezoelectric material for practical applications is its ability to generate and to detect stress waves, i.e. to convert electrical energy into mechanical energy and vice versa. As it is well known, the electromechanical coupling factor, k, fully characterize this energy conversion under static conditions. In a previous work we demonstrated that, like in the static conditions, it is possible to define the k factor in dynamic situations as ratio of energies (k_w); we also showed that k_w is proportional to the static material coupling factor (k_m) of the considered 1-D vibration mode, and the proportionality coefficient does not depend on the mode. In this work we show that the definition of k_w as ratio of energies can also be extended to lossy materials (K_w); the losses are accounted by considering complex quantities for the elastic, dielectric and piezoelectric material constants. The obtained result is that k_w is proportional to the material coupling factor k_m, which in this case is a complex parameter, and the proportionality coefficient is the same of the case without losses.