Ten Issues About Hysteresis
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Ten Issues About Hysteresis Augusto Visintin
Received: 9 December 2013 / Accepted: 25 January 2014 © Springer Science+Business Media Dordrecht 2014
Abstract In this note some basic models of hysteresis are reviewed: the linear and generalized plays, the relay and the Preisach model. Some related questions are also illustrated. Keywords Hysteresis · Ferromagnetism · Elasto-plasticity
1 Introduction This note deals with models of hysteresis phenomena, and expands part of a talk that the author gave at the “Waves and Stability in Continuous Media” (WASCOM) conference, held in Levico in June 2013. These models were already surveyed by this author in a number of works, in particular in [28] and in the recently issued notes [30]. Here the emphasis is on some specific issues, that are briefly discussed after reviewing some basic hysteresis operators. Historical Note Ferromagnetism, elasto-plasticity and some other hysteresis phenomena have been raising the concern of physicists, engineers and other applicative scientists for a long time. However the point of view of functional analysis emerged only recently, first with the pioneering works of Bouc [4, 5] in the 1960s, and then more systematically with the deep and extensive research of the Russian school of M.A. Krasnosel’ski˘ı, A.V. Pokrovski˘ı and co-workers [13, 14] in the 1970s. At the basis of this approach there is the definition of hysteresis as rate-independent memory,1 and the introduction of the notion of (continuous) hysteresis operator acting between Banach spaces of time-dependent functions. Hysteresis in space-distributed systems
1 We shall see that by rate-independence we mean invariance under time rescaling.
In honor of Professor Salvatore Rionero, on the occasion of His 80th birthday
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A. Visintin ( ) Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, 38050 Povo di Trento, Italy e-mail: [email protected]
A. Visintin
was addressed only in the early 1980s. More specifically, continuous hysteresis relations were extended to space-time dependent variables, regarding the space variable as a parameter; these hysteresis models were then coupled with PDEs [21, 22]. At the same time a first weak model was proposed for discontinuous hysteresis, and was coupled with PDEs [23]. This started an extensive research, see e.g. the mathematical monographs [6, 15, 24]. On the more physical and engineering side, ferromagnetic hysteresis was studied e.g. in [1–3, 10, 16, 17]. The coupling of PDEs with continuous hysteresis was extensively addressed, much more than the coupling with discontinuous hysteresis [24]. More recently, a different approach to discontinuous hysteresis emerged for quasi-static evolution (under the key-word of rateindependence, rather than that of hysteresis, although the two terms are essentially equivalent). Rate-independent quasi-static evolution was introduced about fifteen years ago by Francfort and Marigo [11], and was independently rediscovered by Dal Maso and Toader [8], Mielke, Theil, Levitas [19].
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