A thermodynamic approach to rate-type models in deformable ferroelectrics

Abstract

The purpose of the paper is to establish vector-valued rate-type models for the hysteretic properties in deformable ferroelectrics within the framework of continuum thermodynamics. Unlike electroelasticity and piezoelectricity, in ferroelectricity both the polarization and the electric field are simultaneously independent variables so that the constitutive functions depend on both. This viewpoint is naturally related to the fact that an hysteresis loop is a closed curve in the polarization–electric field plane. For the sake of generality, the deformation of the material and the dependence on the temperature are allowed to occur. The constitutive functions are required to be consistent with the principle of objectivity and the second law of thermodynamics. Objectivity implies that the constitutive equations are form invariant within the set of Euclidean frames. Among other results, the second law requires a general property on the relation between the polarization and the electric field via a differential equation. This equation shows a dependence fully characterized by two quantities: the free energy and a function which is related to the dissipative character of the hysteresis. As a consequence, different hysteresis models may have the same free energy. Models compatible with thermodynamics are then determined by appropriate selections of the free energy and of the dissipative part. Correspondingly, major and minor hysteretic loops are plotted.

Thermodynamically Consistent Models of Elastic-Plastic Materials

“…Later on, this assumption was used in [4]. Recently we have shown how the constitutive function for σ is essential in the modelling of ferroelectric hysteresis [5,6]. We now show that the constitutive property of the entropy production allows a thermodynamically consistent class of elastic-plastic models.…”

“…As is standard in continuum mechanics [1,2] we consider any set of constitutive equations as physically admissible if the Clausius-Duhem inequality (5), or (6), holds identically for any process subject to the balance equations ( 1), (2). Equation (3) defines the value of the entropy production; σ > 0 characterizes dissipative phenomena.…”

Magneto-Viscoelastic Materials: Memory Functionals and Rate Equations

“…This is consistent with the view that the dissipation increases proportionally to the friction caused by inter-particle magnetic attraction [ 27 ]. The dual constitutive assumption models the hysteresis effects in ferromagnetic materials, parameterized by the temperature and the strain , as well as with hysteresis in ferroelectrics [ 28 ].…”

A Phase-Field Approach to Continuum Damage Mechanics

“…The thermodynamic consistency is investigated through the constraint of the Clausius–Duhem inequality following the standard procedure of Rational Thermodynamics [ 7 ]. Lately, a more general scheme has been applied in that the entropy production is considered as a constitutive function; this view has proved essential in the elaboration of hysteretic models [ 8 , 9 , 10 ]. Here, the scheme is improved by viewing the entropy production as a sum of two terms, one induced by the other constitutive equations and one given by a constitutive equation of the entropy production per se (extended entropy production).…”