Hysteresis and Phase Transitions pp 22-121 | Cite as

# Hysteresis Operators

## Abstract

Our first approach to the phenomenon of hysteresis is a direct one. We consider the hysteresis diagrams and loops as they present themselves, without trying to understand how their respective forms might follow from general physical principles such as, for example, the universal balance laws for mass, linear momentum and internal energy^{1}; instead, we assume the form of the hysteresis loops as given, and we study them from a purely mathematical point of view. The central notion will be that of a *hysteresis operator*, usually denoted by *W*. Defined in accordance with the rules and functions that accompany a given hysteresis model, a hysteresis operator *W* maps *input functions v = v(t)* into *output functions w = w(t)* (we denote the independent variable by *t* since it will always represent a time variable). Using this formulation, we will examine the *structures* and the resulting *memory effects* of various kinds of hysteresis models. In addition, we establish the relevant connections between the different types of hysteresis operators and basic notions of calculus such as *continuity* and *function spaces*.

### Keywords

Entropy Fatigue Expense Hull Lution## Preview

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### References

- 2.We also refer to the recent collection of lecture notes edited by Visintin (1994b).Google Scholar
- 4.In the literature, this operator is often referred to as the
*Ishlinskii operator*or the*Ishlinshii transducer*, see Krasnoselskii-Pokrovskii (1989). We prefer to name it after Prandtl whose paper appeared more than a decade before Ishlinskii’s who studied this operator in the early forties.Google Scholar - 5.For an account of Preisach’s life, see Vajda-Della Torre (1995).Google Scholar
- 14.See Krasnoselskii-Pokrovskii (1989), Section 34.2.Google Scholar
- 16.See Krasnoselskii-Pokrovskii (1989), Section 2.Google Scholar
- 18.See Krasnoselskii-Pokrovskii (1989), p. 158, in their discussion of
*vector*hysteresis models. Actually, this has only recently been used in the analysis of scalar hysteresis operators.Google Scholar - 19.The estimate (3.21) constitutes the first result published by the group around Krasnoselskii on hysteresis operators, see Krasnoselskii et al. (1970).Google Scholar
- 23.See Krasnoselskii-Pokrovskii (1989), p. 365.Google Scholar
- 27.See Brokate (1989a), Proposition 2.6, and Krejčí-Lovicar (1990), Lemma 2.Google Scholar
- 29.This result is due to Visintin (1994a).Google Scholar
- 30.See Hönig (1975) for an exposition of some basic properties of regulated functions.Google Scholar
- 31.See Krasnoselskii-Pokrovskii (1989), Section 6.6.Google Scholar
- 38.See also Section 2.8 below.Google Scholar
- 40.See Krejčí (1989), Brokate (1989a).Google Scholar
- 44.Cf. (4.23).Google Scholar
- 46.See Brokate-Visintin (1989).Google Scholar
- 47.The generalized play operator serves as the basic hysteresis model in Krasnoselskii-Pokrovskii (1989), where an extensive discussion under minimal assumptions can be found. The connection with the Preisach model has been noted in Krasnoselskii-Pokrovskii (1976).Google Scholar
- 48.See the treatments in Visintin (1994a) and Alt (1985).Google Scholar
- 49.Accordingly, such hysteresis models are often referred to as
*independent domain models*, see Everett et al. (1952, 1954, 1955).Google Scholar - 50.See e.g. Woodward-Delia Torre (1960).Google Scholar
- 52.See, for example, Section II.8 in Mayergoyz (1991).Google Scholar
- 56.See also Brokate et al. (1995c).Google Scholar
- 60.For a different approach, see also Rychlik (1987, 1992, 1993).Google Scholar
- 63.See Matsuishi-Endo (1968).Google Scholar
- 67.See Murakami (1992), McDowell-Ellis (1993), Dreßler et al. (1994).Google Scholar
- 68.After Wöhler (1870).Google Scholar
- 70.Cf. Basquin (1910).Google Scholar
- 71.See Palmgren (1924) and Miner (1945).Google Scholar
- 73.See Krüger et al. (1985).Google Scholar
- 79.See Krejčí (1991a).Google Scholar
- 84.See also Iwan (1966, 1967).Google Scholar
- 85.See Masing (1926).Google Scholar
- 89.This function has been introduced in Everett (1955).Google Scholar
- 90.This feature is often referred to as the
*congruency property*of the Preisach model, see Mayergoyz (1985).Google Scholar - 91.See Remark 2.4.15.Google Scholar
- 92.This result is due to Mayergoyz (1985), see also Brokate (1990a).Google Scholar
- 95.For treatments of the aspects of numerical approximation and identification, we refer the interested reader to the works of Verdi-Visintin (1985,1989), Hoffmann-Sprekels-Visintin (1988), Hoffmann-Meyer (1989), Brokate (1990c), Mayergoyz (1991), Hütter (1991) and Verdi (1994).Google Scholar
- 97.See Mayergoyz-Priedman (1988) and Mayergoyz (1991).Google Scholar
- 100.The original contribution in this connection is Krejčí (1991a), see Remark 2.10.14.Google Scholar
- 102.For the strictly monotone case, part (ii) of the proposition is due to Krejčí (1991a).Google Scholar
- 104.See Proposition 2.2.16.Google Scholar
- 105.See Remark 2.4.15.Google Scholar
- 107.See Krejčí (1991a).Google Scholar
- 108.This result is originally due to Krejčí (1986a).Google Scholar
- 111.See Delia Torre (1966). For recent refinements, we refer to Vajda et al. (1992) and Vajda-Della Torre (1993a, 1993b).Google Scholar
- 113.This result is due to Krejčí (1991a) and Brokate (1992a).Google Scholar
- 114.See Brokate-Visintin (1989).Google Scholar
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- 119.The results of this section are due to Brokate-Dreßler-Krejčí (1995b).Google Scholar