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A thermodynamic approach to hysteretic models in ferroelectrics. (English) Zbl 1510.78025

Summary: The purpose of the paper is to establish a constitutive model for the hysteretic properties in ferroelectrics. Both the polarization and the electric field are simultaneously independent variables so that the constitutive functions depend on both of them. This viewpoint is naturally related to the fact that an hysteresis loop is a closed curve surrounding the region of interest. 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 second law of thermodynamics. Among other results, the second law implies 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 the free energy and a dependence 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.

MSC:

78A48 Composite media; random media in optics and electromagnetic theory
80A19 Diffusive and convective heat and mass transfer, heat flow
82D45 Statistical mechanics of ferroelectrics
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[1] Berti, A.; Giorgi, C.; Vuk, E., Free energies in one-dimensional models of magnetic transitions with hysteresis, Nuovo Cimento Soc. Ital. Fis. B, 125, 371-394 (2010)
[2] Berti, A.; Giorgi, C.; Vuk, E., Hysteresis and temperature-induced transitions in ferromagnetic materials, Appl. Math. Model., 39, 820-837 (2015) · Zbl 1449.78001
[3] Coleman, B. D.; Hodgdon, M. L., A constitutive relation for rate-independent hysteresis in ferromagnetically soft materials, Internat. J. Engrg. Sci., 24, 897-919 (1986) · Zbl 0582.73002
[4] Coleman, B. D.; Hodgdon, M. L., On a class of constitutive relations for ferromagnetic hysteresis, Arch. Ration. Mech. Anal., 99, 375-396 (1987) · Zbl 0631.73093
[5] Devonshire, A. F., Theory of ferroelectrics, Adv. Phys., 3, 85-130 (1954) · Zbl 0056.23901
[6] Fabrizio, M.; Giorgi, C.; Morro, A., A thermodynamic approach to non-isothermal phase-field evolution in continuum physics, Physica D, 214, 144-156 (2006) · Zbl 1096.80002
[7] Fabrizio, M.; Giorgi, C.; Morro, A., Phase transition in ferromagnetism, Internat. J. Engrg. Sci., 47, 821-839 (2009) · Zbl 1213.78014
[8] Francois-Lavet, V.; Henrotte, F.; Stainier, L.; Noels, L., An energy-based variational model of ferromagnetic hysteresis for finite element computations, J. Comput. Appl. Math., 246, 243-250 (2013) · Zbl 1267.78003
[9] Gentili, G.; Giorgi, C., A new model for rate-independent hysteresis in permanent magnets, Internat. J. Engrg. Sci., 39, 1057-1090 (2001) · Zbl 1210.78008
[10] B. Jayawardhana, R. Ouyang, V. Andrieu, Dissipativity of general Duhem hysteresis models, in: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC, New York, 2011, pp. 3234-3239.
[11] Kovetz, A., Electromagnetic Theory, 69 (2000), Oxford Scienc Pub.: Oxford Scienc Pub. Oxford · Zbl 1038.78001
[12] Landau, L. D.; Lifshitz, E. M.; Pitaevskii, L. P., Statistical Physics (1980), Pergamon Press: Pergamon Press Oxford
[13] Marton, P.; Rychetsky, I.; Hlinka, J., Domain walls of ferroelectric BaTiO_3 within the Ginzburg-Landau-Devonshire phenomenological model, Phys. Rev. B, 81, Article 144125 pp. (2010)
[14] Moulson, A. J.; Herbert, J. M., Electroceramics, 339-410 (2003), Wiley: Wiley New York
[15] Müller, I., The coldness a universal function in thermoelastic bodies, Arch. Ration. Mech. Anal., 41, 319-332 (1971) · Zbl 0225.73003
[16] Pao, Y.-S.; Hutter, K., Electrodynamics for moving elastic solids and viscous fluids, Proc. IEEE, 63, 1011-1021 (1975)
[17] Sarjala, M.; Seppälä, E. T.; Alava, M. J., Dynamic hysteresis in ferroelectrics with quenched randomness, Physica B, 403, 418-421 (2008)
[18] Su, Y.; Landis, C. M., Continuum thermodynamics of ferroelectric domain evolution: theory, finite element implementation, and application to domain wall pinning, J. Mech. Phys. Solids, 55, 280-305 (2007) · Zbl 1419.82074
[19] Tiersten, H. F., On the nonlinear equations of thermoelectroelasticity, Internat. J. Engrg. Sci., 9, 587-604 (1971) · Zbl 0225.73082
[20] Visintin, A., Differential Models of Hysteresis (1994), Springer: Springer Berlin · Zbl 0820.35004
[21] Visintin, A., Mathematical models of hysteresis, (Bertotti, G.; Mayergoyz, I., The Science of Hysteresis (2006), Elsevier), 1-123 · Zbl 1149.35077
[22] Vörös, J., Modeling and identification of hysteresis using special forms of the Coleman-Hodgdon model, J. Electr. Eng., 60, 100-105 (2009)
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