© 2015

Rate-Independent Systems

Theory and Application

  • The mathematical framework for a rigorous mathematical treatment of rate-independent systems, together with various applications, is presented in a comprehensive form for the first time.

  • Provides existence theorems and tools for discussing numerical solutions.

  • Key international researchers in this field Motivates mathematical theory for scientists by use of examples


Part of the Applied Mathematical Sciences book series (AMS, volume 193)

Table of contents

  1. Front Matter
    Pages i-xxii
  2. Alexander Mielke, Tomàš Roubíček
    Pages 1-43
  3. Alexander Mielke, Tomàš Roubíček
    Pages 45-115
  4. Alexander Mielke, Tomàš Roubíček
    Pages 117-234
  5. Alexander Mielke, Tomàš Roubíček
    Pages 235-458
  6. Alexander Mielke, Tomàš Roubíček
    Pages 459-577
  7. Back Matter
    Pages 579-660

About this book


This monograph provides both an introduction to and a thorough exposition of the theory of rate-independent systems, which the authors have worked on with a number of collaborators over many years. The focus is mostly on fully rate-independent systems, first on an abstract level with or without a linear structure, discussing various concepts of solutions with full mathematical rigor. The usefulness of the abstract concepts is then demonstrated on the level of various applications primarily in continuum mechanics of solids, including suitable approximation strategies with guaranteed numerical stability and convergence. Particular applications concern inelastic processes such as plasticity, damage, phase transformations, or adhesive-type contacts both at small strains and at finite strains. Other physical systems such as magnetic or ferroelectric materials, and couplings to rate-dependent thermodynamic models are also considered. Selected applications are accompanied by numerical simulations illustrating both the models and the efficiency of computational algorithms.

This book presents the mathematical framework for a rigorous mathematical treatment of rate-independent systems in a comprehensive form for the first time. Researchers and graduate students in applied mathematics, engineering, and computational physics will find this timely and well-written book useful.


computational mechanics contact mechanics continuum mechanics of solids damage dissipation potentials energetic solutions energy functionals evolutionary gamma convergence inelastic processes mutual recovery sequences numerical methods for rate-independent systems partial differential inequalities phase transformations plasticity rate-independent evolutionary systems thermodynamics weak solutions

Authors and affiliations

  1. 1.Weierstraß-Institut fürAngewandte Analysis und StochastikBerlinGermany
  2. 2.Charles UniversityMathematical InstitutePragueCzech Republic

About the authors

Alexander Mielke is Professor at Humboldt University of Berlin as well as a head of the research group "Partial Differential Equations" at the Weierstrass Institute for Applied Analysis and Stochastics. His research interest range over applied mathematical analysis, partial differential equations, multi-scale modeling and applications in continuum physics.

Tomáš Roubíček is a Professor at Charles University in Prague, as well as a researcher at Institute of Thermomechanics and also at Institute of Information Theory and Automation of the Czech Academy of Sciences. Having an engineering background, his professional activity has evolved from computer simulations of systems of nonlinear partial differential equations thru numerical mathematics and optimization theory to applied mathematical analysis focused to mathematical modeling in engineering and physics.

Bibliographic information

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“The book is an interesting and useful piece of applied mathematics. It is of definite interest for researchers involved in the mathematical issues of the models in materials sciences. The book contains a clear treatment of the mathematical methods we need and a survey of many applications.” (Giuseppe Saccomands, Mathematical Reviews, July, 2016)

“The book is written by two famous specialists of evolutionary systems as they worked for a long time on this topic. Throughout the whole book, the authors give lots of precise details concerning the particularities of the rate-independent systems in different and realistic settings. They take care to illustrate their theoretical notions with many examples and they complete their computations with many figures.” (Alain Brillard, zbMATH 1339.35006, 2016)

“This monograph presents the research of more than one decade by the authors and their collaborators. It is strongly recommended for all scientists in the field of modeling and analyzing mechanical systems and may establish as a basic reference for further developments.” (Christian Wieners, ZAMM, Zeitschrift fur Angewandte Mathematik und Mechanik, Vol. 96 (4), 2016)