Abstract
In the previous chapters, the occurrence of hysteresis has been discussed from a mainly phenomenological and mathematical point of view. The attempt to formalize the input-output behaviour of hysteresis loops gave rise to introduce the notion of hysteresis operators, and we studied their properties and differential equations in which they appear. In this approach, we did not pay much attention to the physical circumstances. In particular, we entirely ignored the fact that in nature hysteresis effects are often caused by phase transitions which are accompanied by abrupt changes of some of the involved physical quantities, as well as by the absorption or release of energy in the form of latent heat. The area of the hysteresis loop itself gives a measure for the amount of energy that has been dissipated or absorbed during the phase transformation.
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References
For a more detailed treatment of the theory of phase transitions, we refer the reader to the monographs of Stanley (1971) and Ma (1976).
See Devonshire (1954).
In particular, the second principle of thermodynamics and the so-called principle of material frame indifference impose strong restrictions on the form that the constitutive equations may have. For a detailed discussion, we refer to Müller (1985).
For the definition of these quantities, we refer to Müller (1985).
For a detailed treatment of the Landau approach to temperature-induced first-order phase transitions, we refer to Chapter IV in Tolédano-Tolédano (1987).
Cf. Devonshire (1954).
In this respect we refer to the works of Modica (1987), Owen (1989) and Fonseca-Tartar (1989), for instance.
See Cahn-Allen (1977).
This equation was first considered in Cahn-Hilliard (1958).
See Remark 6.4.4 at the end of Chapter 6.
Actually, in Alt-Pawlow (1990,1992a) a more general form has been assumed for both the fluxes j and q; see also de Groot-Mazur (1984).
Other examples to which this theory applies are the lattice gas or the metallic order-disorder transition, see Penrose-Fife (1990).
Cf. Caginalp (1986,1989,1991), Caginalp-Lin (1987) and Caginalp-Fife (1988), for non-conserving dynamics, and Caginalp (1990), for conserving dynamics.
Problems of this type are usually referred to as free boundary problems, with the unknown interface Γ(t) playing the role of the free boundary. For surveys on the existing mathematical theory of free boundary problems, we refer the reader to Fasano-Primicerio (1983a, 1983b), Bossavit-Damlamian-Frémond (1985a, 1985b), Hoffmann-Sprekels (1990a, 1990b) and Chadam-Rasmussen (1993a, 1993b, 1993c).
Cf. Gibbs (1948).
Recently, some of these formal arguments have been made rigorous for certain cases; in this respect, we refer the reader to Stoth (1996a, 1996b) and Soner (1994).
For the explicit form of the constant c 0, we refer to Caginalp (1990).
For recent results, see Luckhaus (1990), Chen-Reitich (1992) and Soner (1994).
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© 1996 Springer-Verlag New York, Inc.
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Brokate, M., Sprekels, J. (1996). Phase Transitions and Hysteresis. In: Hysteresis and Phase Transitions. Applied Mathematical Sciences, vol 121. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4048-8_5
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DOI: https://doi.org/10.1007/978-1-4612-4048-8_5
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