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2015 | Buch

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Rate-Independent Systems

Theory and Application

verfasst von: Alexander Mielke, Tomáš Roubíček

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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 accompanied by 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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. A general view of rate-independent systems

Abstract

This chapter provides a basic introduction to the concepts and notions developed in this book. We begin from the perspective of ordinary differential equations arising in mechanics

Alexander Mielke, Tomàš Roubíček

Chapter 2. Energetic rate-independent systems

Abstract

To keep the connection with continuum mechanics, cf. also Section 1.3.2, we consider the basic state space split to two spaces

$$\displaystyle\begin{array}{rcl} \mathcal{Q} = \mathcal{Y}\times \mathcal{Z},& &{}\end{array}$$

(2.0.1)

where the fast component y and the slow component z of the state q = (y, z) live. Whenever possible, however, we will write q instead of (y, z) to shorten the notation. The splitting is done such that the evolution of z in time involves dissipation, whereas that of y does not. The state space $\mathcal{Q}$ is equipped with a Hausdorff topology $\mathcal{T}_{\mathcal{Q}} = \mathcal{T}_{\mathcal{Y}}\times \mathcal{T}_{\mathcal{Z}}$, and we denote by $q_{k}^{}\stackrel{\!\mathcal{Q}}{\rightarrow }q$, $y_{k}^{}\stackrel{\!\mathcal{Y}}{\rightarrow }y$, and $z_{k}^{}\stackrel{\!\mathcal{Z}}{\rightarrow }z$ the corresponding convergence of sequences. Throughout, it will be sufficient to consider sequential closedness, compactness, and continuity. For notational convenience, we will not write this explicitly.

Alexander Mielke, Tomàš Roubíček

Chapter 3. Rate-independent systems in Banach spaces

In the Banach-space setting, we assume that the topologies are given by either the weak or the strong topology. Throughout this chapter, we will assume that the topological spaces $\mathcal{Y}$ and $\mathcal{Z}$ of Chapter 2 are given via separable reflexive Banach spaces $\boldsymbol{Y }$ and $\boldsymbol{Z}$ equipped with their weak topologies, unless stated otherwise explicitly. Thus $\mathcal{Y}\subset \boldsymbol{Y }$, $\mathcal{Z}\subset \boldsymbol{Z}$, and $\mathcal{Q}:= \mathcal{Y}\times \mathcal{Z}$ from (2.0.1) is a subset of

$$\displaystyle\begin{array}{rcl} \boldsymbol{Q} = \boldsymbol{Y } \times \boldsymbol{Z}\quad \mbox{ with separable, reflexive Banach spaces $\boldsymbol{Y }$ and $\boldsymbol{Z}$.}& &{}\end{array}$$

(3.0.1)

In Banach spaces, we have two important additional tools deriving from the linear structure. First, the functionals at hand may have differentials or subdifferentials such that it is possible to formulate force balances, such as

$$\displaystyle{ \partial _{\dot{q}}\mathcal{R}(q(t),\dot{q}(t)) + \partial _{q}\mathcal{E}(t,q(t)) \ni 0\qquad \mbox{ for a.a. }\ t\! \in \! [0,T], }$$

(3.0.2)

and to formulate rate equations rather than compare energies, as in the energetic formulation. Second, we can employ convexity and duality methods such as the Legendre–Fenchel transform, as indicated in Section 1.3.4 Here we use the symbol ∂ for the Fréchet subdifferential (cf. Section 3.3.1 for the definition), which generalizes the convex subdifferential and the Fréchet derivative, and $\partial _{a}\mathcal{J} (a,b)$ or $\partial _{b}\mathcal{J} (a,b)$ denotes the partial Fréchet subdifferentials, where b or a is kept fixed, respectively.

Alexander Mielke, Tomàš Roubíček

Chapter 4. Applications in continuum mechanics and physics of solids

Abstract

The theory of rate-independent processes has a large variety of applications in continuum mechanics of solids. Rate-independent effects typically can occur inside the bulk and at the surface or along interfaces. These effects may be unidirectional, as, for example, in damage, or bidirectional. In case of a deformable continuum, one can consider the general concept of large strains or confine oneself to small strains. There might be rate-independent processes on lower-dimensional objects, typically surfaces of codimension 1 or lines (as dislocations) of dimension 1. See Table 4.1 on p. 236 for examples that will be considered in this chapter. Of course, various processes can combine with each other.

Alexander Mielke, Tomàš Roubíček

Chapter 5. Beyond rate-independence

Abstract

Sometimes, dynamical systems host various processes, only some of which are rate-independent. These processes can be manifested depending on loading regimes, and only in some regimes (typically very slow) do the rate-independent processes dominate; cf. also Sect. 5.1.2.2 below. In other regimes, however, the modeling assumption about rate-independence is inapplicable.

Alexander Mielke, Tomàš Roubíček

Backmatter

Titel: Rate-Independent Systems
verfasst von: Alexander Mielke
Tomáš Roubíček
Verlag: Springer New York
Electronic ISBN: 978-1-4939-2706-7
Print ISBN: 978-1-4939-2705-0
DOI: https://doi.org/10.1007/978-1-4939-2706-7

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