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Finite Element Formulation for Ferroelectric Hysteresis of Piezoelectric Materials

For the numerical simulation of non-linear piezoelectric material behavior, we use a constitutive relation that is based on a decomposition of the physical quantities dielectric displacement and mechanical strain into a reversible and an irreversible part. Therein, we set the irreversible part of the dielectric displacement equal to the irreversible electric polarization and express the irreversible mechanical strain by a polynomial ansatz of the irreversible electric polarization. The reversible parts of mechanical strain and dielectric displacement are further described by the linear piezoelectric constitutive law. We apply a Preisach hysteresis operator to compute the irreversible polarization from the history of the driving electric field. Furthermore, the entries of the piezoelectric modulus tensor are assumed to be functions of the electric polarization. To efficiently solve the non-linear system of partial differential equations, we have developed a quasi-Newton scheme and use the finite element (FE) method for the numerical solution. This FE scheme has been applied to numerically calculate the dynamic behavior of a piezoelectric disc and a stack actuator. The obtained results compare well to measured data.