Hysteresis and Phase Transitions

Front Cover
Springer Science & Business Media, Jun 20, 1996 - Mathematics - 358 pages
Hysteresis is an exciting and mathematically challenging phenomenon that oc curs in rather different situations: jt, can be a byproduct offundamental physical mechanisms (such as phase transitions) or the consequence of a degradation or imperfection (like the play in a mechanical system), or it is built deliberately into a system in order to monitor its behaviour, as in the case of the heat control via thermostats. The delicate interplay between memory effects and the occurrence of hys teresis loops has the effect that hysteresis is a genuinely nonlinear phenomenon which is usually non-smooth and thus not easy to treat mathematically. Hence it was only in the early seventies that the group of Russian scientists around M. A. Krasnoselskii initiated a systematic mathematical investigation of the phenomenon of hysteresis which culminated in the fundamental monograph Krasnoselskii-Pokrovskii (1983). In the meantime, many mathematicians have contributed to the mathematical theory, and the important monographs of 1. Mayergoyz (1991) and A. Visintin (1994a) have appeared. We came into contact with the notion of hysteresis around the year 1980.
 

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Contents

Some Mathematical Tools
10
11 Measure and Integration
11
12 Function Spaces
14
13 Nonlinear Equations
18
14 Ordinary Differential Equations
20
Hysteresis Operators
22
21 Basic Examples
23
22 General Hysteresis Operators
32
Phase Transitions and Hysteresis
150
41 Thermodynamic Notions and Relations
151
42 Phase Transitions and Order Parameters
154
43 Landau and Devonshire Free Energies
156
44 Ginzburg Theory and Phase Field Models
163
Hysteresis Effects in Shape Memory Alloys
175
52 WellPosedness for Falks Model
181
53 Numerical Approximation
204

23 The Play Operator
42
24 Hysteresis Operators of Preisach Type
52
25 Hysteresis Potentials and Energy Dissipation
66
26 Hysteresis Counting and Damage
71
27 Characterization of Preisach Type Operators
80
28 Hysteresis Loops in the Prandt1 Model
86
29 Hysteresis Loops in the Preisach Model
93
210 Composition of Preisach Type Operators
99
211 Inverse and Implicit Hysteresis Operators
105
212 Hysteresis Count and Damage Part II
117
Hysteresis and Differential Equations
122
31 Hysteresis in Ordinary Differential Equations
124
32 Auxiliary Imbedding Results
126
33 The Heat Equation with Hysteresis
128
34 A Convexity Inequality
138
35 The Wave Equation with Hysteresis
140
54 Complementary Remarks
215
Phase Field Models With NonConserving Kinetics
218
61 Auxiliary Results from Linear Elliptic and Parabolic Theory
219
62 WellPosedness of the Caginalp Model
227
63 WellPosedness of the PenroseFife Model
242
64 Complementary Remarks
267
Phase Field Models With Conserved Order Parameters
271
71 WellPosedness of the Caginalp Model
274
72 WellPosedness of the PenroseFife Model
283
Phase Transitions in Eutectoid Carbon Steels
304
82 The Mathematical Model
307
83 WellPosedness of the Model
311
A Numerical Study
329
Bibliography
332
Index
353
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