# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

August 2015 , Volume 8 , Issue 4

Issue on rate-independent evolutions and hysteresis modelling

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2015, 8(4): i-i doi: 10.3934/dcdss.2015.8.4i +[Abstract](1762) +[PDF](91.3KB)
Abstract:
The interest in hysteresis and rate-independent phenomena is shared by scientists with a great variety of different backgrounds. We can encounter these processes in several situations of common life: for instance in elasto-plasticity, ferromagnetism, shape-memory alloys, phase transitions. Beyond physics, hysteresis and rate-independent phenomena appear also in engineering, biology, economics as well as in many other settings, playing an important role in many applications. The complexity arising in these fields necessarily requires a joint contribution of experts with different backgrounds and skills. Therefore, only synergy and cooperation among these several people can lead to concrete advances in the technological capabilities of our society.
This special issue of Discrete and Continuous Dynamical Systems is devoted to the latest advances and trends in the modelling and in the analysis of this family of complex phenomena. In particular, we gathered contributions from different fields of science (mathematical analysis, mathematical physics, engineering) with the intent of presenting an updated picture of current research directions, offering a new and interdisciplinary perspective in the study of these processes.
Motivated by the Spring School on Rate-independent Evolutions and Hysteresis Modelling, held at the Politecnico di Milano and University of Milano on May 27-31, 2013, this special issue contains different kinds of original contributions: some of them originate from the courses held in that occasion and from the discussions they stimulated, but are here presented in a new perspective; some others instead are original contributions in related topics. All the papers are written in the clearest possible language, accessible also to students and non-experts of the field, with the intent to attract and introduce them to this topic.
Final acceptance of all the papers in this volume was made by the normal referee procedure and standard practices of AIMS journals.
We wish to thanks all the referees, who kindly agreed to devote their time and effort to read and check all the papers carefully, providing useful comments and recommendations. We are also grateful to all the authors for their great job and the high quality of their contributions. We finally wish to express our gratitude to AIMS and in particular to Prof. Alain Miranville for the opportunity to publish this special issue and for the technical support.
2015, 8(4): 649-691 doi: 10.3934/dcdss.2015.8.649 +[Abstract](1463) +[PDF](1951.9KB)
Abstract:
These notes origin from a group of lectures given at the Spring School on Rate-independent evolutions and hysteresis modelling'' (Hystry 2013), held at Politecnico di Milano and at Università degli Studi di Milano, from May 27 until May 31, 2013. They are addressed to Graduate students in mathematics and applied science, interested in modeling rate-independent effects in smart systems. Therefore, they aim to provide the basic issues concerning modeling of multi-functional materials showing memory phenomena, with emphasis to magnetostrictives, in view of their application to the design of smart devices. Such tutorial summarizes several years activity on these issues that involved the cooperation with several colleagues, among all Dr. P. Krejčí, with whom the authors are indebted.
2015, 8(4): 693-722 doi: 10.3934/dcdss.2015.8.693 +[Abstract](1225) +[PDF](2879.5KB)
Abstract:
Non-isothermal phase-field models of transition phenomena in materials with hysteresis are considered within the framework of the Ginzburg-Landau theory. Our attempt is to capture the relation between phase-transition and hysteresis (either mechanical or magnetic). All models are required to be compatible with thermodynamics and to fit well the shape of the major hysteresis loop. Focusing on uniform cyclic processes, numerical simulations at different temperatures are performed.
2015, 8(4): 723-747 doi: 10.3934/dcdss.2015.8.723 +[Abstract](1693) +[PDF](584.8KB)
Abstract:
Shape-memory alloys are active materials, their amazing thermo-electromechanical behavior is at the basis of a variety of innovative applications. Many models have been set forth in order to describe this complex behavior. Among these the so-called Souza-Auricchio model appears as remarkably simple in terms of mechanical assumptions yet accurate in the description of three-dimensional experiments and robust with respect to approximations. Our aim is to survey here the current literature on the Souza-Auricchio model, with a specific focus on modeling.
2015, 8(4): 749-756 doi: 10.3934/dcdss.2015.8.749 +[Abstract](1187) +[PDF](314.7KB)
Abstract:
We discuss two inverse problems of reconstruction of data in a mixed parabolic integrodifferential problem. First, we shall consider the reconstruction on a factor depending on time in the source term. Next, we shall consider the reconstruction of a convolution kernel.
2015, 8(4): 757-767 doi: 10.3934/dcdss.2015.8.757 +[Abstract](1014) +[PDF](322.2KB)
Abstract:
The goal of this note is to discuss the basic thermodynamical principles and show how they need to be considered in the process of developing new mathematical models. We give numerous examples: linear elasticity with constant or non-constant temperature, we discuss classical hysteresis models as the play operator, the Preisach operator as well as new models introduced in the last years - the temperature dependent Preisach model, models of magnetostriction and models of an oscillating beam with fatigue.
2015, 8(4): 769-772 doi: 10.3934/dcdss.2015.8.769 +[Abstract](1095) +[PDF](257.2KB)
Abstract:
We review the proof of existence and uniqueness of the Poisson's equation $\Delta u + {\rm div}\,{\bf m}=0$ whenever ${\bf m}$ is a unit $L^2$-vector field on $\mathbb R^3$ with compact support; by standard linear potential theory we deduce also the $H^1$-regularity of the unique weak solution.
2015, 8(4): 773-792 doi: 10.3934/dcdss.2015.8.773 +[Abstract](1476) +[PDF](474.3KB)
Abstract:
Motivated by the sweeping processes, we develop an abstract theory of continuous hysteresis operators acting between rectifiable curves with values in metric spaces. In particular we study the continuity properties of such operators and how they can be extended from the space of Lipschitz continuous functions to the space of rectifiable curves. Applications to the sweeping processes and to the vector play operator are shown.
2015, 8(4): 793-816 doi: 10.3934/dcdss.2015.8.793 +[Abstract](1490) +[PDF](604.3KB)
Abstract:
Continuous and discontinuous hysteresis operators are first reviewed in general. The Duhem model, the generalized play, the (delayed) relay and the Preisach model are outlined, as well as vector extensions of the two latter models.
Two examples of initial- and boundary-value problems for P.D.E.s with hysteresis are then illustrated. Well-posedness is proved for quasilinear parabolic problems with either continuous or discontinuous hysteresis. Existence of a weak solution is shown for second-order quasilinear hyperbolic problems with discontinuous hysteresis.

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