## Hysteresis and Phase TransitionsHysteresis 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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 |

353 | |

### Other editions - View all

### Common terms and phrases

addition analysis apply assertion assume assumptions Banach space boundary conditions bounded called Chapter conclude Consequently consider constant continuous converges corresponding curve defined definition deletion denote density depend derivative determined discrete dx ds dynamics equation estimates example existence fact field Finally finite fixed follows free energy function given heat Hence holds hysteresis operator implies increasing inequality initial input integrate interval introduce Lemma linear mapping mathematical measure memory Moreover multiply nonlinear Note numerical obtain order parameter particular phase transitions physical piecewise monotone play positive Prandtl operator Preisach operator Proof properties Proposition proved rate independent reader recall refer remains Remark respectively result rule satisfying side solution space string sufficiently Suppose temperature Theorem theory thermodynamic transformation unique yields Young's inequality