Thin piezolectric rods with quasi-static domain switching

Following an increasing use of piezoelectric materials to design "smart" sensor-actuator systems, a lot of attention is paid to the modeling of their electromechanical beahviour, as testified in literature by many survey articles and books; however, since most of the applications deal with linear plate-like or rod-like bodies, they consequently embody the same kind of approximate theories that are used in technical mechanics, with further ad-hoc assumptions on the electromechanical coupling. In order to remove these degrees of approximations, we obtained, from the three-dimensional theory a model for the one-dimensional dynamics of linear piezoelectric rods without introducing an "order of magnitude" of kinematical terms which are regarded to be small: this was achieved by using the method of internal constraints, which reflects at a phenomenological level the concept of mechanical thinness of the rod, whereas the electrical thinness was reflected on a semi-inverse assumption on the electric displacement field. Such a simple model for the rod electromechanical motions showed both the relation between electric field and deformation as well as the piezoelectric stiffening effect. Here we wish to get the same results by taking into account the domain switching phenomena: we limit our analysis to the quasi-static case and use in place of the standard Voigt's linear relations, those proposed by P.J. Chen. Once we get from this macroscopic constitutive model and from our hypothesis an expression for the electric and the displacement fields, by using standard variational techniques we arrive at the one-dimensional equations of motion and rate-law.