G. in [4,6,7]. In the next section we illustrate a classic model of vector ferromagnetic hysteresis, that deals with a finer length-scale than that the Preisach model is supposed to represent. 5 MICROMAGNETISM In this section we outline the theory of micromagnetism, and introduce the Landau-Lifshitz equation. In order to account for dissipation due to magnetic inclusions, we then amend this equation, and derive a purely rateindependent mesoscopic model of ferromagnetic hysteresis. Spontaneous Magnetization Even a small magnetic field is capable of inducing a large magnetization in a ferromagnetic body below a temperature (named Curie's temperature) that is characteristic of the material.
Denoting series and parallel arrangements respectively by the symbols ' - ' and '1', this model can be represented by the rheological formula E - P; cf. Fig. 5. 43), denoting by e the total displacement and by a the (signed) force intensity, one easily derives the following variational inequality lalKb, de a dt da)(adt v)~O u Ivl~b. 1 (18; Chapter II]). , a ~ is continuous in W1,1(0, T), and can be extended to a Lipschitz-continuous operator in C O([0, T]). Alternatively, one may first define ~ for piecewise linear inputs, check that this operator is Lipschitz-continuous with respect to the uniform metric, and then extend it to C O([0, T]) [1,3,5,8,16].
The Science of Hysteresis : 3-volume set by Giorgio Bertotti, Isaak D. Mayergoyz