Table Of ContentApplied Mathematical Sciences
Volume 121
Editors
IE. Marsden L. Sirovich F. John (deceased)
Advisors
M. Ohil IK. Hale T. Kambe
J. Keller K. Kirchgiissner
B.J. Matkowsky C.S. Peskin
J.T. Stuart
Springer
New York
Berlin
Heidelberg
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Santa Clara
Singapore
Tokyo
Applied Mathematical Sciences
I. John: Partial Differential Equations, 4th ed. 34. KevorkianiCole: Perturbation Methods in
2. Sirovich: Techniques of Asymptotic Analysis. Applied Mathematics.
3. Hale: Theory of Functional Differential 35. Carr: Applications of Centre Manifold Theory.
Equations, 2nd ed. 36. Bengtsson/GhillKiillen: Dynamic Meteorology:
4. Percus: Combinatorial Methods. Data Assimilation Methods.
5. von Mises/Friedrichs: Fluid Dynamics. 37. Saperstone: Semidynamical Systems in Infinite
6. Freiberger/Grenander: A Short Course in Dimensional Spaces.
Computational Probability and Statistics. 38. Lichtenberg/Lieberman: Regular and Chaotic
7. Pipkin: Lectures on Viscoelasticity Theory. Dynamics, 2nd ed.
8. Giacoglia: Perturbation Methods in Non-linear 39. Piccini/Stampacchia/Vidossich: Ordinary
Systems. Differential Equations in R".
9. Friedrichs: Spectral Theory of Operators in 40. Naylor/Sell: Linear Operator Theory in
Hilbert Space. Engineering and Science.
10. Stroud: Numerical Quadrature and Solution of 41. Sparrow: The Lorenz Equations: Bifurcations,
Ordinary Differential Equations. Chaos, and Strange Attractors.
II. Wolovich: Linear Multivariable Systems. 42. Guckenheimer/Holmes: Nonlinear Oscillations,
12. Berkovitz: Optimal Control Theory. Dynamical Systems and Bifurcations of Vector
13. Bluman/Cole: Similarity Methods for Fields.
Differential Equations. 43. OckendoniTaylor: Inviscid Fluid Flows.
14. Yoshizawa: Stability Theory and the Existence 44. PaZ)/: Semigroups of Linear Operators and
of Periodic Solution and Almost Periodic Applications to Partial Differential Equations.
Solutions. 45. GlashoffiGustafson: Linear Operations and
15. Braun: Differential Equations and Their Approximation: An Introduction to the
Applications, 3rd ed. Theoretical Analysis and Numerical Treatment
16. Lefschetz: Applications of Algebraic Topology. of Semi-Infinite Programs.
17. CollatziWetterling: Optimization Problems. 46. Wilcox: Scattering Theory for Diffraction
18. Grenander: Pattern Synthesis: Lectures in Gratings.
Pattern Theory, Vol. I. 47. Hale et al: An Introduction to Infinite
19. Marsden/McCracken: Hopf Bifurcation and Its Dimensional Dynamical Systems-Geometric
Applications. Theory.
20. Driver: Ordinary and Delay Differential 48. Murray: Asymptotic Analysis.
Equations. 49. Ladyzhenskaya: The Boundary-Value Problems
21. Courant/Friedrichs: Supersonic Flow and Shock of Mathematical Physics.
Waves. 50. Wilcox: Sound Propagation in Stratified Fluids.
22. RouchelHabets/Laloy: Stability Theory by 51. Golubitsky/Schaeffer: Bifurcation and Groups in
Liapunov's Direct Method. Bifurcation Theory, Vol. I.
23. Lamperti: Stochastic Processes: A Survey of the 52. Chipot: Variational Inequalities and Flow in
Mathematical Theory. Porous Media.
24. Grenander: Pattern Analysis: Lectures in Pattern 53. Majda: Compressible Fluid Flow and System of
Theory, Vol. II. Conservation Laws in Several Space Variables.
25. Davies: Integral Transforms and Their 54. Wasow: Linear Turning Point Theory.
Applications, 2nd ed. 55. Yosida: Operational Calculus: A Theory of
26. Kushner/Clark: Stochastic Approximation Hyperfunctions.
Methods for Constrained and Unconstrained 56. Chang/Howes: Nonlinear Singular Perturbation
Systems. Phenomena: Theory and Applications.
27. de Boor: A Practical Guide to Splines. 57. Reinhardt: Analysis of Approximation Methods
28. Keilson: Markov Chain Models-Rarity and for Differential and Integral Equations.
Exponentiality. 58. Dwoyer/HussainilVoigt (eds): Theoretical
29. de Veubeke: A Course in Elasticity. Approaches to Turbulence.
30. Shiatycki: Geometric Quantization and Quantum 59. Sanders/Verhulst: Averaging Methods in
Mechanics. Nonlinear Dynamical Systems.
31. Reid: Sturmian Theory for Ordinary Differential 60. GhiliChildress: Topics in Geophysical
Equations. Dynamics: Atmospheric Dynamics, Dynamo
32. Meis/Markowitz: Numerical Solution of Partial Theory and Climate Dynamics.
Differential Equations.
33. Grenander: Regular Structures: Lectures in
Pattern Theory, Vol. III. (continued following index)
Martin Brokate Ji irgen Sprekels
Hysteresis and
Phase Transitions
With 43 Illustrations
i
Springer
M. Brokate J. Sprekels
Mathematisches Seminar Weierstrass Institute for Applied
Christian-Albrechts-Universitat zu Kiel Analysis and Stochastics
D-24098 Kiel Mohrenstrasse 39
Germany D-I0117 Berlin
Germany
Editors
IE. Marsden L. Sirovich
Division of Applied Mathematics
Control and Dynamical Systems, 104-44
Brown University
California Institute of Technology
Providence, RI 02912
Pasadena, CA 91125
USA
USA
Mathematics Subject Classification (1991): 35K60, 35L50, 35Q99, 47H30, 49S05, 65M60,
73B05,73E05,80A22
Library of Congress Cataloging-in-Publication Data
Brokate, Martin, 1953-
Hysteresis and phase transitions I Martin Brokate, J iirgen
Sprekels.
p. cm.-(Applied mathematical sciences; 121)
Includes bibliographical references. '
ISBN-13: 978-1-4612-8478-9 e-ISBN-13: 978-1-4612-4048-8
001: 10.1007/978-1-4612-4048-8
1. Hysteresis - Mathematics. 2. Phase transformations (Statistical
physics) I. Sprekels, J. II. Title. III. Series: Applied
mathematical sciences (Springer-Verlag New York Inc.) ; v. 121.
QA1.A647 no. 121
[QC754.2.H9)
510 s-dc20
[530.1 '557248) 96-15533
Printed on acid-free paper.
© 1996 Springer-Verlag New York, Inc.
Softcover reprint of the hadcover 1s t edition 1996
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New
York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereaf
ter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even
if the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
by anyone.
Production managed by Karina Gershkovich; manufacturing supervised by Jacqui Ashri.
Photocomposed pages prepared from the author's U'lEX file using Springer-Verlag's
"svsing.sty" macro.
987654321
Dedicated to our families
Preface
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.
During that period, the second author investigated the control of heating sys
tems via thermostat relaysl. Later, after we had both become members of a
group around K.-H. Hoffmann at the University of Augsburg that investigated
free boundary problems, we learned during a workshop in Heidelberg about
A. Visintin's fundamental results on the heat equation with hysteresis. Ever
since, we have devoted a large part of our own research to the investigation of
the phenomenon of hysteresis, extending our studies to phase transitions and
their thermodynamic foundations when, in 1984, H. W. Alt, 1. Muller and M.
Niezg6dka brought us into contact with the exciting interplay between the hys
teresis effects and the accompanying austenite-martensite phase transitions in
shape memory alloys.
Much of the material covered in this volume is original and resulted from our
studies when we were affiliated with the Universities of Augsburg, Essen, Kai
serslautern, Kiel, with the Humboldt-University of Berlin and the Weierstrass
Institute for Applied Analysis and Stochastics in Berlin.
This monograph is primarily addressed to applied mathematicians. We do
hope, however, that some of its material will prove useful also for scientists from
the applied fields in which hysteresis occurs, such as physics, materials science,
lCf. Glashoff-Sprekels (1981,1982).
vii
viii Preface
chemistry and engineering.
During the preparation of this book, we obtained much encouragement and
many helpful hints from a number of colleagues. We thank H. W. Alt, A. Fried
man, K.-H. Hoffmann, P. KrejCi, M. Niezg6dka, A. Visintin and S. Zheng for
countless inspiring discussions, and we express our special gratitude to I. Muller
and K. Wilmanski for their continuing readiness to discuss experimental and
thermodynamical questions connected with hysteresis. We are also indebted to
Springer-Verlag, especially to Dr. J. Heinze, for their continuing encouragement
during the preparation of this monograph.
Finally, we would like to thank Mr. J. Sieber, who produced the figures
in this book, and Mrs. J. Lohse, for improving the English of the text. The
LaTeX-setting of the text has been done by the authors themselves; therefore,
we have the full responsibility for each occasional misprint in this monograph.
Kiel and Berlin, November 1995
M. Brokate and J. Sprekels
Contents
Preface ... Vll
Introduction 1
Chapter 1. Some Mathematical Tools 10
1.1 Measure and Integration . 11
1.2 Function Spaces . . . . . 14
1.3 Nonlinear Equations . . . 18
1.4 Ordinary Differential Equations . 20
Chapter 2. Hysteresis Operators 22
2.1 Basic Examples ..... . 23
2.2 General Hysteresis Operators 32
2.3 The Play Operator . . . . . . 42
2.4 Hysteresis Operators of Preisach Type 52
2.5 Hysteresis Potentials and Energy Dissipation 66
2.6 Hysteresis Counting and Damage . . . . . . . 71
2.7 Characterization of Preisach Type Operators 80
2.8 Hysteresis Loops in the Prandtl Model . . 86
2.9 Hysteresis Loops in the Preisach Model . 93
2.10 Composition of Preisach Type Operators 99
2.11 Inverse and Implicit Hysteresis Operators 105
2.12 Hysteresis Count and Damage, Part II .. 117
Chapter 3. Hysteresis and Differential Equations 122
3.1 Hysteresis in Ordinary Differential Equations 124
3.2 Auxiliary Imbedding Results .... 126
3.3 The Heat Equation with Hysteresis . 128
3.4 A Convexity Inequality . . . . . . . 138
3.5 The Wave Equation with Hysteresis 140
IX
x Contents
Chapter 4. Phase Thansitions and Hysteresis .. 150
4.1 Thermodynamic Notions and Relations . 151
4.2 Phase Thansitions and Order Parameters . 154
4.3 Landau and Devonshire Free Energies . . 156
4.4 Ginzburg Theory and Phase Field Models 163
Chapter 5. Hysteresis Effects in Shape Memory Alloys 175
5.1 Phenomenology and Falk's Model. 175
5.2 Well-Posedness for Falk's Model 181
5.3 Numerical Approximation ..... 204
5.4 Complementary Remarks . . . . . 215
Chapter 6. Phase Field Models With Non-Conserving Kinetics 218
6.1 Auxiliary Results from Linear Elliptic and Parabolic Theory . 219
6.2 Well-Posedness of the Caginalp Model . . . 227
6.3 Well-Posedness of the Penrose-Fife Model . 242
6.4 Complementary Remarks . . . . . . . . . . 267
Chapter 7. Phase Field Models With Conserved Order Parameters 271
7.1 Well-Posedness of the Caginalp Model . . . . . . . 274
7.2 Well-Posedness of the Penrose-Fife Model. . . . . 283
Chapter 8. Phase Thansitions in Eutectoid Carbon Steels. 304
8.1 Phenomenology of the Phase Thansitions . 304
8.2 The Mathematical Model . . . . . . . 307
8.3 Well-Posedness of the Model . . . . . 311
8.4 The Jominy Test: A Numerical Study 329
Bibliography 332
Index .... 353
Introduction
When speaking of hysteresis 1 , one usually refers to a relation between two scalar
time-dependent quantities that cannot be expressed in terms of a single-valued
function, but takes the form of loops like the one depicted in Fig. 0.1.
P
D
Fig. 0.1. Typical schematic elastoplastic response in a load-deformation cycle.
In Fig. 0.1, the output of a basic experiment in material science is sketched: a
cylindrical specimen of some material is subjected to a time-varying axial load
P(t) , while its deformation D(t) is measured.
By definition, elastic behaviour occurs within some range [p-, P+l, if the
relation between load and deformation can be expressed by a function f as
D(t) = f(P(t)) , (0.1)
for all loading histories satisfying P(t) E [p-, P+l for all times t.
Plastic behaviour is more complicated: imagine a tensile load that increases
from 0 to some (sufficiently large) value Po > 0, followed by unloading back to
0; afterwards, a compressive load is applied up to -Po, followed by unloading
to 0 and, finally, by a tensile loading to Po. Then the typical elastoplastic
1 From the Greek word hysterein = to be behind or later, to come late; hyster
esis = shortcoming, deficiency, need. See Liddell-Scott-Jones (1843/1951).
1
