Three-dimensional nonlocal models of deformable ferroelectrics: A thermodynamically consistent approach

Within the framework of continuum thermodynamics, the paper develops a scheme for the study of piezo-ferroelectric materials undergoing large deformations. The ferroelectric polarization is decomposed into the sum of a reversible and a residual part which is con- sidered as an independent variable. The modeling of the constitutive properties is made simpler by using referential, Euclidean invariant quantities. Constitutive functions depend on a set of variables that includes their time derivatives and gradients, and therefore must be consistent with a nonlocal statement of the second law of thermodynamics where the entropy production is represented as the sum of a non-negative supply and a flux. Both terms are also assigned by means of constitutive functions. A new and quite general model relating the residual-polarization rate and the ‘electric Gibbs free entropy’ is established. This thermodynamic potential is modeled according to the Landau-Devonshire approach and appropriate explicit expressions are proposed for anisotropic materials. Consequently, a new explicit evolution equation for the polarization vector is obtained for both high and low temperatures. In particular, the Landau-Devonshire scalar model is recovered in the isotropic case. Under the assumption of small strains and small polarization gradients, hys- teresis and electromechanical coupling are described for a simple one-dimensional model. The advantage of our original approach is that a few constitutive parameters provide a good fit of material behavior.