##
**Hysteresis, convexity and dissipation in hyperbolic equations.**
*(English)*
Zbl 1187.35003

GAKUTO International Series. Mathematical Sciences and Applications 8. Tokyo: Gakkotosho (ISBN 4-7625-0417-3). vii, 211 p. (1996).

Hysteresis is a common feature of many physical, mechanical, natural, economical phenomena such as, for instance: ferromagnetism, plasticity in continuum dynamics, filtration through porous media, behaviour of thermostats, biological dynamics, profit-investment strategies. From an analytical or an applicative point of view, sometimes it is a “bad” feature, it makes the system more complex, it prevents the possibility of exactly calculating the state of the system (think for instance to models of damage), sometimes it is a “good” feature in the sense that an artificial hysteresis is introduced both in the theoretical and in the real model in order to prevent some undesired behaviours such as, for instance, fast oscillations of the dynamics (think, for instance, to a thermostat).

As definition, we can say that hysteresis is an input/output relationship between two time-dependent quantities which are suitable to describe the state of the system (constitutive law), for instance: magnetic field/magnetization in ferromagnetism, stress/strain in continuum dynamics, pressure/saturation in filtration through porous media. In particular, we say that we are in the presence of hysteresis if such a relationship presents a particular kind of memory. This means that the state of the output at a fixed instant depends on the whole past history of the input and not only on the state of the input at the same instant; moreover such a memory should be “rate independent”, that is, roughly speaking, it should be independent on the velocity (time-derivative) of the input in describing its history, but it should only depend on the sequence of values reached by the input. This particular feature enables us to represent the input/output relationship in a phase-space where commonly the characteristic loops (hysteresis cycles) appear.

From an analytical point of view, the hysteretic relationship can be described by means of some functional operators (namely: hysteresis operators) which act on some suitable spaces of time-dependent functions. Several examples of such operators exist, each one of them particularly indicated for some applications. All of them are nonlinear and very often non-monotone (at least in the usual sense as operator between Banach spaces), and this fact makes hard their study, in particular in connection with partial differential equations. However, the rate independence memory and the possible convexity of the hysteresis loops, are sometimes helpful to overcome the lacking of linearity and monotonicity. This is indeed one of the main interesting feature of the present book.

The book under review, which was written in the last century by one of the leading researchers on the field, certainly constitutes a milestone on the theory of hyperbolic equations with hsyteresis features, where the convexity of the hysteresis loops plays a crucial and somehow new role (linked to the dissipation of energy).

Citing from the Preface: “The present volume is mainly devoted to the mathematical aspects of rate independent plastic hysteresis in continuum dymanics. 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”.

“…although the (quasilinear) equation of motion with hysteretic constitutive law preserves its hyperbolicity characterized by the finite speed of propagation, it can be solved considerably more easily than quasilinear equations without hysteresis by the methods of semilinear equations”.

“This book is intended to give a consistent and self-contained presentation of the theory and its connection with other disciplines. In Chapter I we interpret hysteresis within the classical approach to continuum mechanics and derive analytical properties of hsyteresis operators arising from rheological models. The efficiency of the hysteresis 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 of fatigue and damage”.

“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 situations. Chapter V is an appendix, where we try to incorporate specific auxiliary functional-analytical results into a larger theory in order to make them more accessible to the reader”.

Here is the list of contents. I. Hysteresis operators in mechanics. I.1 Rheological models: composition of rheological elements, kinematic hardening, isotropic and kinematic hardening, multiyield models; I.2 Geometry of convex sets: recession cone, tangent and normal cone, strict convexity; I.3 The play and stop operators: continuous inputs, regularity, periodic inputs, energy inequalities; I.4 Special characteristics: cylinders, strictly convex cylinders, polyhedrons, smooth characteristics.

II. Scalar models of hysteresis. II.1 Scalar play and stop: Lipschitz continuity, output variation; II.2 Memory of the play-stop system: differentiability, monotonicity; II.3 Multiyield scalar hysteresis models: Prandtl-Ishlinskii operators, Preisach operator, the Della Torre model; II.4 Monotonicity and energy inequalities: thermodynamical consistency, monotonicity, second order energy inequalities; II.5 Models of fatigue and damage: a nonlinear elasto-plasto-brittle model, a differential model of fatigue.

III. Hyperbolic equations with hysteretic constitutive laws. III.1 Construction of solutions: monotonicity method, compactness method; III.2 Uniqueness and asymptotics: uniqueness, nonresonance, decay of solutions; III.3 Periodic solutions: compactness method, monotonicity method, asymptotic method.

IV. The Riemann problems. IV.1 Weak self-similar solutions: non-existence of smooth solutions, weak soloutions, multiplicity of weak solutions; IV.2 Dissipation of energy: dissipation condition, multiplicity of dissipative solutions; IV.3 Minimal solutions: monotone solutions, existence and uniqueness of minimal solutions, existence and uniqueness in the Riemann problems; IV.4 Entropy conditions: Lax entropy condition, Liu’s shock admissibility criterion, Dafermos maximal entropy rate criterion, vanishing viscosity.

V. Appendix: Function spaces. V.1 Integration of vector-valued functions: Bochner integral, functions of bounded variations, Stieltjes integral; V.2 Embedding theorems

As definition, we can say that hysteresis is an input/output relationship between two time-dependent quantities which are suitable to describe the state of the system (constitutive law), for instance: magnetic field/magnetization in ferromagnetism, stress/strain in continuum dynamics, pressure/saturation in filtration through porous media. In particular, we say that we are in the presence of hysteresis if such a relationship presents a particular kind of memory. This means that the state of the output at a fixed instant depends on the whole past history of the input and not only on the state of the input at the same instant; moreover such a memory should be “rate independent”, that is, roughly speaking, it should be independent on the velocity (time-derivative) of the input in describing its history, but it should only depend on the sequence of values reached by the input. This particular feature enables us to represent the input/output relationship in a phase-space where commonly the characteristic loops (hysteresis cycles) appear.

From an analytical point of view, the hysteretic relationship can be described by means of some functional operators (namely: hysteresis operators) which act on some suitable spaces of time-dependent functions. Several examples of such operators exist, each one of them particularly indicated for some applications. All of them are nonlinear and very often non-monotone (at least in the usual sense as operator between Banach spaces), and this fact makes hard their study, in particular in connection with partial differential equations. However, the rate independence memory and the possible convexity of the hysteresis loops, are sometimes helpful to overcome the lacking of linearity and monotonicity. This is indeed one of the main interesting feature of the present book.

The book under review, which was written in the last century by one of the leading researchers on the field, certainly constitutes a milestone on the theory of hyperbolic equations with hsyteresis features, where the convexity of the hysteresis loops plays a crucial and somehow new role (linked to the dissipation of energy).

Citing from the Preface: “The present volume is mainly devoted to the mathematical aspects of rate independent plastic hysteresis in continuum dymanics. 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”.

“…although the (quasilinear) equation of motion with hysteretic constitutive law preserves its hyperbolicity characterized by the finite speed of propagation, it can be solved considerably more easily than quasilinear equations without hysteresis by the methods of semilinear equations”.

“This book is intended to give a consistent and self-contained presentation of the theory and its connection with other disciplines. In Chapter I we interpret hysteresis within the classical approach to continuum mechanics and derive analytical properties of hsyteresis operators arising from rheological models. The efficiency of the hysteresis 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 of fatigue and damage”.

“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 situations. Chapter V is an appendix, where we try to incorporate specific auxiliary functional-analytical results into a larger theory in order to make them more accessible to the reader”.

Here is the list of contents. I. Hysteresis operators in mechanics. I.1 Rheological models: composition of rheological elements, kinematic hardening, isotropic and kinematic hardening, multiyield models; I.2 Geometry of convex sets: recession cone, tangent and normal cone, strict convexity; I.3 The play and stop operators: continuous inputs, regularity, periodic inputs, energy inequalities; I.4 Special characteristics: cylinders, strictly convex cylinders, polyhedrons, smooth characteristics.

II. Scalar models of hysteresis. II.1 Scalar play and stop: Lipschitz continuity, output variation; II.2 Memory of the play-stop system: differentiability, monotonicity; II.3 Multiyield scalar hysteresis models: Prandtl-Ishlinskii operators, Preisach operator, the Della Torre model; II.4 Monotonicity and energy inequalities: thermodynamical consistency, monotonicity, second order energy inequalities; II.5 Models of fatigue and damage: a nonlinear elasto-plasto-brittle model, a differential model of fatigue.

III. Hyperbolic equations with hysteretic constitutive laws. III.1 Construction of solutions: monotonicity method, compactness method; III.2 Uniqueness and asymptotics: uniqueness, nonresonance, decay of solutions; III.3 Periodic solutions: compactness method, monotonicity method, asymptotic method.

IV. The Riemann problems. IV.1 Weak self-similar solutions: non-existence of smooth solutions, weak soloutions, multiplicity of weak solutions; IV.2 Dissipation of energy: dissipation condition, multiplicity of dissipative solutions; IV.3 Minimal solutions: monotone solutions, existence and uniqueness of minimal solutions, existence and uniqueness in the Riemann problems; IV.4 Entropy conditions: Lax entropy condition, Liu’s shock admissibility criterion, Dafermos maximal entropy rate criterion, vanishing viscosity.

V. Appendix: Function spaces. V.1 Integration of vector-valued functions: Bochner integral, functions of bounded variations, Stieltjes integral; V.2 Embedding theorems

Reviewer: Fabio Bagagiolo (Trento)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

35L70 | Second-order nonlinear hyperbolic equations |

47J40 | Equations with nonlinear hysteresis operators |