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.

1) Phase Transitions, represented by generalizations of the classical Stefan problem. This is studied by Kenmochi and Rodrigues by means of variational techniques. 2) Hysteresis Phenomena. Some alloys exhibit shape memory effects, corresponding to a stress-strain relation which strongly depends on temperature; mathematical physical aspects are treated in Müller's paper. In a general framework, hysteresis can be described by means of hysteresis operators in Banach spaces of time dependent functions; their properties are studied by Brokate. 3) Numerical analysis. Several models of the phenomena above can be formulated in terms of nonlinear parabolic equations. Here Verdi deals with the most updated approximation techniques.

Hysteresis phenomena are common in numerous physical, mechanical, ecological and biological systems. They reflect memory effects and process irreversibility. The use of hysteresis operators (hysterons) offers an approach to macroscopic modelling of the dynamics of phase transitions and rheological systems. The applications cover processes in electromagnetism, elastoplasticity and population dynamics in particular. Hysterons are also typical elements of control systems where they represent thermostats and other discontinuous controllers with memory. The book offers the first systematic mathematical treatment of hysteresis nonlinearities. Construction procedures are set up for hysterons in various function spaces, in continuous and discontinuous cases. A general theory of variable hysterons is developed, including identification and stability questions. Both deterministic and non-deterministic hysterons are considered, with applications to the study of feedback systems. Many of the results presented - mostly obtained by the authors and their scientific group - have not been published before. The book is essentially self contained and is addressed both to researchers and advanced students.

This 2006 work began with the author's exploration of the applicability of the finite deformation theory of elasticity when various standard assumptions such as convexity of various energies or ellipticity of the field equations of equilibrium are relinquished. The finite deformation theory of elasticity turns out to be a natural vehicle for the study of phase transitions in solids where thermal effects can be neglected. This text will be of interest to those interested in the development and application of continuum-mechanical models that describe the macroscopic response of materials capable of undergoing stress- or temperature-induced transitions between two solid phases. The focus is on the evolution of phase transitions which may be either dynamic or quasi-static, controlled by a kinetic relation which in the framework of classical thermomechanics represents information that is supplementary to the usual balance principles and constitutive laws of conventional theory.

This book deals with the phenomenological theory of first-order structural phase transitions, with a special emphasis on reconstructive transformations in which a group-subgroup relationship between the symmetries of the phases is absent. It starts with a unified presentation of the current approach to first-order phase transitions, using the more recent results of the Landau theory of phase transitions and of the theory of singularities. A general theory of reconstructive phase transitions is then formulated, in which the structures surrounding a transition are expressed in terms of density-waves, providing a natural definition of the transition order-parameters, and a description of the corresponding phase diagrams and relevant physical properties. The applicability of the theory is illustrated by a large number of concrete examples pertaining to the various classes of reconstructive transitions: allotropic transformations of the elements, displacive and order-disorder transformations in metals, alloys and related structures, crystal-quasicrystal transformations.

The use of high-pressure techniques has become popular for studying the nature of substances and phenomena occurring in them, especially as a means of obtaining new materials (synthesis under high pressure) and processing known materials (hydroextrusion). A product of many years of research by the authors and their colleagues, Phase Transitions in Solids under High Pressure discusses the relationships of phase transformations in solids under high pressure, the mechanism of these transformations, crystal geometry, the effect of deformation, the conditions of formation, and preservation of the high-pressure phases under normal pressure. The book begins with an introduction that describes the relationship of the thermodynamics of phase transformations and the kinetics of the transformations. This is followed by a chapter explaining the equipment and mostly original procedures for investigating phase transformation in solids under high hydrostatic and quasi-hydrostatic pressures. The book covers phase transformations under high pressure in a wide temperature range in the elements carbon, silicon, germanium, titanium, zirconium, iron, gallium, and cerium as well as in titanium- and iron-based alloys and AIBVII, AIIBVI, and AIIIBV compounds. In addition, the book examines the kinetics of phase transformations in iron-based alloys in isobaric–isothermal conditions. The authors present results for phase transformations in deformation under high pressure, describe several non-trivial effects associated with phase transformations under high pressure, and analyze the kinetics and hysteresis of high-temperature and low-temperature phase transformations. They conclude by describing the role of investigations under high pressure for determining general relationships governing phase transformations in solids.

A new class of insulating solids was recently discovered. Whenirradiated by a few visible photons, these solids give rise to amacroscopic excited domain that has new structural and electronicorders quite different from the starting ground state. This occurrenceis called photoinduced phase transition, and this multi-authoredbook reviews recent theoretical and experimental studies of this newphenomenon.

The author's experimental discoveries in the field of solid-state phase transitions have brought about a thorough explanation of this phenomenon, including the puzzling nature of "lamda-anomalies." These phase transitions are found to be always a nucleation and crystal growth in a solid medium, while "second (or higher) order" phase transitions are a misconception: they do not exist. Ramifications of this new understanding are substatial. In this book the reader will find the first unified account for fundamentals of the three great areas of solid-state physics? Phase transitions, ferromagnetism and ferroelectricity, free of the inconsistencies of the conventional theories.