**Contents.**

Preface;

I. Hysteresis operators in mechanics;

II. Scalar models of hysteresis;

III. Hyperbolic equations with hysteretic constitutive laws;

IV. The Riemann problem;

V. Appendix: function spaces.

**Preface.**

It is not necessary to make a long introduction in order to justify that the mathematical theory of hysteresis gives a useful tool for solving concrete engineering problems in various branches of applied research. A sufficient evidence is presented in the monographs that recently appeared or will appear in the near future (Krasnosel'skii and Pokrovskii (1983), Mayergoyz (1991), Visintin (1994), Brokate and Sprekels (1996)) which cover a broad area of the theory and applications.

The present volume is mainly devoted to mathematical aspects of rate independent plastic hysteresis in continuum dynamics. The results of Chapters II and III can however be interpreted also in the framework of Maxwell's equations in ferromagnetic media of Preisach or Della Torre type. In any case, coupling hysteretic constitutive laws with the equations of motion we are led to quasilinear hyperbolic equations with hysteretic terms. This is a completely new branch of applied mathematics at the early stage where, following Hrych (1991), one can say with not so much exaggeration that "fabrication is the most reliable reference".

The situation is very different here from the theory of parabolic equations with hysteresis developed by Visintin in the 80's (see Visintin (1994)) which is an extension (sometimes very nontrivial) of the ideas and techniques derived from the general theory of quasilinear parabolic equations and applied to specific hysteretic nonlinearities. This is by no means the case of hyperbolic equations with hysteresis and the conclusion is surprising: although the (quasilinear) equation of motion with a hysteretic constitutive law preserves its hyperbolicity characterized by the finite speed of propagation, it can be solved considerably more easily than quasilinear hyperbolic equations without hysteresis by the methods of semilinear equations.

There is no simple and satisfactory explanation of this fact. We nevertheless make here a comparison of the behavior of solutions to one-dimensional quasilinear wave equations with and without hysteresis. While the latter develop discontinuities (shocks) in a finite time and weak solutions are not uniquely determined, so that additional physically motivated conditions have to be prescribed, hysteresis constitutive operators with convex loops in the former case exhibit a higher order energy dissipation which enables us to derive strong a priori estimates and pass to the limit in a suitable approximation scheme. From the geometrical point of view, if we represent the solutions of the Riemann problem for the equation without hysteresis by their trajectories in the strain - stress diagram, then shocks correspond to straight segments connecting two points on the constitutive graph. We observe that shocks are always organized in such a way that the corresponding trajectory is convex if the solution increases and concave if it decreases. The maximal dissipation principle then selects the solution with the minimal convex/maximal concave trajectory. We can say that some kind of spontaneous hysteresis occurs even if no hysteresis is assumed in the constitutive law itself. If now the constitutive law is given by a hysteresis operator with convex loops, it is natural to expect that the solution will follow smoothly their convex/concave branches and shocks have no reason to occur.

There are other interesting coincidences which would merit deeper understanding. This is for instance the question of the role of the two maximal dissipation principles in the rigid - plastic constitutive law (Sect. I.1) and in the Riemann problem (Sect. IV.3) which are in some sense responsible for the generation of hysteresis. We also do not comment on the fact that the Preisach operator itself is governed by a hyperbolic equation, where the memory variable plays the role of time (Sect. II.3).

This book is intended to give a consistent and self-contained presentation of the theory and its connection to other disciplines. In Chapter I we interpret hysteresis within the classical approach to continuum mechanics and derive analytical properties of hysteresis operators arising from rheological models. The efficiency of the hysteretic description depends on the complexity of the memory structure. In Chapter II we study the memory induced by scalar hysteresis models of Prandtl - Ishlinskii, Preisach, Della Torre and two models for fatigue and damage.

The main and rather nontrivial feature of hysteresis operators consists in the fact that they dissipate energy of two orders which relate to the area of closed hysteresis loops and to the curvature of their branches, respectively. We derive corresponding energy inequalities which enable us subsequently in Chapter III to construct solutions to hyperbolic equations with hysteretic constitutive laws. Chapter IV gives a detailed study of the Riemann problem with a not necessarily monotone nonlinearity without hysteresis and shows how hysteresis appears in the physically relevant solutions. Chapter V is an appendix, where we try to incorporate specific auxiliary functional-analytic results into a larger theory in order to make them more accessible to the reader.

Statements and formulae in the text are numbered consecutively in each section. References to results from other chapters are preceded by the roman number of the chapter. Thus, for example, Proposition I.3.9 refers to Proposition 3.9 of Chapter I, equation (3.26) means the corresponding formula in the chapter where the reference is made etc.

The author is indebted to Professor Otto Vejvoda, Vladimir Lovicar and Ivan Straskraba from Prague, Pierre-Alexandre Bliman from Paris, Martin Brokate from Kiel and Augusto Visintin from Trento for stimulating discussions and encouragement. The final redaction of the manuscript was made possible thanks to Dasa Berkova and Karel Horak from the Mathematical Institute of the Academy of Sciences of the Czech Republic.

Pavel Krejčí

Prague, January 1995