Overview
- 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
- Includes supplementary material: sn.pub/extras
Part of the book series: Applied Mathematical Sciences (AMS, volume 655)
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Table of contents (5 chapters)
Keywords
- 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
- partial differential equations
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 been working on with a lot of collaborators over 15 years. The focus is mostly on fully rate-independent systems, first on an abstract level either with or even without a linear structure, discussing various concepts of solutions with full mathematical rigor. Then, usefulness of the abstract concepts is 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. A few other physical systems, e.g. magnetic or ferroelectric materials, and couplings to rate-dependent thermodynamic models are considered as well. Selected applications are accompaniedby numerical simulations illustrating both the models and the efficiency of computational algorithms.
In this book, the mathematical framework for a rigorous mathematical treatment of "rate-independent systems" is presented 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.
Reviews
“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)
Authors and Affiliations
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
Book Title: Rate-Independent Systems
Book Subtitle: Theory and Application
Authors: Alexander Mielke, Tomáš Roubíček
Series Title: Applied Mathematical Sciences
DOI: https://doi.org/10.1007/978-1-4939-2706-7
Publisher: Springer New York, NY
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Science+Business Media New York 2015
Hardcover ISBN: 978-1-4939-2705-0Published: 21 October 2015
Softcover ISBN: 978-1-4939-4783-6Published: 23 August 2016
eBook ISBN: 978-1-4939-2706-7Published: 21 October 2015
Series ISSN: 0066-5452
Series E-ISSN: 2196-968X
Edition Number: 1
Number of Pages: XXII, 660
Number of Illustrations: 62 b/w illustrations, 13 illustrations in colour
Topics: Partial Differential Equations, Mathematical Methods in Physics, Solid Mechanics