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1996 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Hysteresis and Phase Transitions

verfasst von: Martin Brokate, Jürgen Sprekels

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
When speaking of hysteresis1, 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.
Martin Brokate, Jürgen Sprekels
Chapter 1. Some Mathematical Tools
Abstract
In this section, we will collect some material which is standard and covered in many textbooks, in order to facilitate the reading of this monograph. We restrict ourselves to only a few basic mathematical tools, assuming that the reader has a working knowledge of the calculus of one and several real variables and is familiar with the basic notions of linear functional analysis. In particular, we make free use of such fundamental concepts as Banach spaces, Hilbert spaces and dual spaces, compactness, strong convergence, weak convergence and weak-star convergence, respectively. Further basic material will be presented when needed in the later chapters of this volume.
Martin Brokate, Jürgen Sprekels
Chapter 2. 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 energy1; 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.
Martin Brokate, Jürgen Sprekels
Chapter 3. Hysteresis and Differential Equations
Abstract
Many dynamical systems exhibit hysteresis as one of their features. In classical continuum mechanics, hysteretic behaviour is inherent in many constitutive laws. In systems and control applications, hysteresis regularly appears via mechanical play and friction, or in the form of a relay or thermostat, often deliberately built into the system. If the hysteretic behaviour is described using a hysteresis operator, then the mathematical model for the dynamical system consists of a system of differential equations coupled with one or several hysteresis operators, which is complemented by initial and boundary conditions. The oscillator with hysteretic restoring force,
$$ y''(t) + W[y](t) = f(t), $$
W being a hysteresis operator, furnishes a basic example1 2.
Martin Brokate, Jürgen Sprekels
Chapter 4. Phase Transitions and Hysteresis
Abstract
In the previous chapters, the occurrence of hysteresis has been discussed from a mainly phenomenological and mathematical point of view. The attempt to formalize the input-output behaviour of hysteresis loops gave rise to introduce the notion of hysteresis operators, and we studied their properties and differential equations in which they appear. In this approach, we did not pay much attention to the physical circumstances. In particular, we entirely ignored the fact that in nature hysteresis effects are often caused by phase transitions which are accompanied by abrupt changes of some of the involved physical quantities, as well as by the absorption or release of energy in the form of latent heat. The area of the hysteresis loop itself gives a measure for the amount of energy that has been dissipated or absorbed during the phase transformation.
Martin Brokate, Jürgen Sprekels
Chapter 5. Hysteresis Effects in Shape Memory Alloys
Abstract
In this chapter, we study the hysteresis effects that are characteristic for the so-called shape memory alloys. Among these materials, there are metallic alloys like Cu Zn, Cu Zn Al, Au Cu Zn, Cu Al Ni and Ni Ti. The hysteretic behaviour of the load-deformation curves for these alloys is accompanied by the so-called shape memory effect which has been exploited in various technological applications.
Martin Brokate, Jürgen Sprekels
Chapter 6. Phase Field Models with Non-Conserving Kinetics
Abstract
In this chapter, we present a mathematical analysis of the dynamics of phase transitions with a non-conserved order parameter e (Model A phase transitions). In this connection, we refer to Section 4.4, where a mathematical model for such phase transitions has been developed that explains the possible occurrence of an accompanying hysteresis in the framework of Landau-Ginzburg theory. The approach of Section 4.4 led to
$$ \frac{{\partial e}}{{\partial t}} = K(e,T)\;\left( {div(\frac{{\gamma (e,T)\nabla e}}{T}) - \frac{1}{{2T}}\;\frac{{\partial \gamma }}{{\partial e}}(e,T)\;|\nabla e{|^2} - \frac{1}{T}\;\frac{{\partial F}}{{\partial e}}(e,T)} \right) $$
(0.1)
as the kinetic equation governing the evolution of the order parameter e, while the balance of internal energy was given by
$$ \frac{\partial }{{\partial t}}\left( {F(e,T) - T\frac{{\partial F}}{{\partial T}}(e,T) + \frac{1}{2}(\gamma (e,T) - T\frac{{\partial \gamma }}{{\partial T}}(e,T))|\nabla e{|^2}} \right) + div\left( {k(e,T)\nabla (\frac{1}{T})} \right) = g $$
(0.2)
.
Martin Brokate, Jürgen Sprekels
Chapter 7. Phase Field Models With Conserved Order Parameters
Abstract
In this chapter, we present a mathematical analysis of the dynamics of phase transitions with a conserved order parameter e (Model B phase transitions). Again, we refer to Section 4.4, where we introduced a mathematical model for Model B phase transitions that explains the possible occurrence of an accompanying hysteresis in the framework of Landau-Ginzburg theory. The kinetic equation for the order parameter e resulting from this approach is given by
$$ \frac{{\partial e}}{{\partial t}}\quad = \quad div\left( {\tilde{K}(e,T)\nabla \left( { - div\left( {\frac{{\gamma (e,T)\nabla e}}{T}} \right) + \frac{1}{{2T}}\frac{{\partial \gamma }}{{\partial e}}(e,T)|\nabla e{|^2} + \frac{1}{T}\frac{{\partial F}}{{\partial e}}(e,T)} \right)} \right) $$
(0.1)
while the balance of internal energy becomes
$$ \frac{\partial }{{\partial t}}\left( {F(e,T) - T\frac{{\partial F}}{{\partial T}}(e,T) + \frac{1}{2}\left( {\gamma (e,T) - T\frac{{\partial \gamma }}{{\partial T}}(e,T)} \right)|\nabla e{|^2}} \right) + div\left( {K(e,T)\nabla \left( {\frac{1}{T}} \right)} \right) = g $$
(0.2)
.
Martin Brokate, Jürgen Sprekels
Chapter 8. Phase Transitions in Eutectoid Carbon Steels
Abstract
In this chapter, we study the austenite-pearlite and austenite-martensite phase changes occurring during the cooling process of a carbon steel of 0.8 per cent carbon content. Such steels are called eutectoid. Since we will investigate cooling processes only in this chapter, hysteresis phenomena will not play such a dominant role as before; however, hysteretic effects come immediately into play if a series of alternating cooling and heating periods is applied to the sample.
Martin Brokate, Jürgen Sprekels
Backmatter
Metadaten
Titel
Hysteresis and Phase Transitions
verfasst von
Martin Brokate
Jürgen Sprekels
Copyright-Jahr
1996
Verlag
Springer New York
Electronic ISBN
978-1-4612-4048-8
Print ISBN
978-1-4612-8478-9
DOI
https://doi.org/10.1007/978-1-4612-4048-8