The rotationally invariant ferro-electroelastic equations describing hysteretic effects in polarized ferroelectric ceramics are obtained from an existing system of electro-elastic equations by employing internal variables in a thermodynamic state function. It is assumed that all material irreversibility is a consequence of the well-known polarization-electric field irreversibility, which is caused by domain switching. Since in polarized ferro-electric ceramic plates which is the configuration commonly employed in applications, the residual polarization is in the fixed thickness direction, only one internal variable is required in the description, namely the polarization in the thickness direction. Tn the usual way, in order to assure that the rate of entropy production is positive, we obtain an evolution equation on the thickness component of polarization. However, since we are interested in slow hysteresis, we do not consider it purposeful to pursue the evolution equation. Instead, we ignore the evolution equation and take the irreversible thickness polarization- electric field curve as the known factor, which is determined from the assumptions qualitatively motivated by the fundamental cause of essentially static irreversibility, i.e., domain switching.; There are 12 new material coefficients in the three-dimensional description and 7 new plate coefficients for the plane-stress case. Besides, there are 3 material coefficients in the usual thickness polarization-electric field saturation curve. Since experimental data is obtained from a thin beam forced into slow motion by the application of voltages applied to actuators on the top and bottom surfaces of the beam, the constitutive equation for the bending moment in the thin beam with the actuators attached is the relation from which the coefficients describing the irreversiblity are determined from one set of experimental data and then used to calculate hysteretic behavior for comparison with other sets of experimental data. The resulting calculations are in good agreement with the measurements. Analysis procedures for flexible structures embedded/attached with actuators are also provided for the purpose of practical applications.
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