A thermodynamic approach to rate-type models in deformable ferroelectrics
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O R I G I NA L A RT I C L E
Claudio Giorgi
· Angelo Morro
A thermodynamic approach to rate-type models in deformable ferroelectrics
Received: 12 June 2020 / Accepted: 4 November 2020 © The Author(s) 2020
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. Keywords Deformable ferroelectrics · Rate-type models · Thermodynamics · Gibbs free energy · Hysteretic function · Major and minor hysteretic loops 1 Introduction Ferroelectrics belong to a large class of materials that exhibit a spontaneous alignment of dipoles which is ascribed to their mutual interaction. All polar crystal classes show a spontaneous polarization without applying a mechanical load and in the absence of an external electric field. Such a behaviour is due to a non-vanishing electric dipole moment associated with their unit cell and depends on temperature. Polar crystals such as poled piezoelectric ceramics exhibit both piezoelectricity and pyroelectricity. The direct piezoelectric effect is the change of polarization when applying a mechanical stress, whereas in the converse piezoelectric effect the application of an electrical field creates mechanical deformation in the crystal. Pyroelectricity can be described as the ability to generate a temporary electric potential in response to a temperature change. A recent review Communicated by Andreas Öchsner. C. Giorgi (B) DICATAM, Università degli studi di Brescia, Via Valotti 9, 25133 Brescia, Italy E-mail: [email protected] A. Morro DIBRIS, University of Genoa, Via All’Opera Pia 13, 16145 Genoa, Italy E-mail: [email protected]
C. Giorgi, A. Morro
of mathematical theories appropriate
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