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2015 | OriginalPaper | Buchkapitel

1. A general view of rate-independent systems

verfasst von : Alexander Mielke, Tomàš Roubíček

Erschienen in: Rate-Independent Systems

Verlag: Springer New York

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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

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Nächstes Kapitel Energetic rate-independent systems
Fußnoten
1
See., e.g., [455, Sect. 15.1.1] for this terminology.
 
2
Special situations can be handled as in [367], where the case α = 2 and \([-\lambda, 0]\) instead of \((-\infty, 0]\) is considered.
 
3
In fact, for d = 2, using in a nontrivial way a sophisticated regularity argument by Gröger [239], the analysis has been performed by Knees [303]; more precisely, even the vectorial situation has been addressed in [303] using [266]. However, for d = 3, this fails if α ≥ 2.
 
4
Historically, internal variables have also been called “internal coordinates” or “internal degrees of freedom”; cf. [72, Ch. 6].
 
5
First, the prefix “pseudo” (as used in [382]) refers to the fact that the actual dissipation rate \(\mathfrak{R}\) is different from the potential \(\mathcal{R}\), in contrast to the “static” energy \(\langle \frac{1} {2}Kq -\ell (t),q\rangle\), which serves equally for the potential of conservative forces and for the energy stored. Yet \(\mathcal{R}\) is simultaneously the dissipation rate and the dissipation potential if and only if it is 1-homogeneous (cf. Proposition 3.​2.​4 for the “only if” part); thus we can legally call \(\mathcal{R}\) a potential. Second, “pseudo” sometimes refers to the nondifferentiability of \(\mathcal{R}\) at 0 (cf. [455]), so that in this terminology, \(\mathcal{R}\) would remain a “pseudopotential” even in the 1-homogeneous case on which this book focuses.
 
6
Usually, the term “flow” refers to the solution semigroup generated by the differential inclusion or equation. If this equation has the above gradient structure, then this semigroup is called a (generalized) gradient flow. We adopt a convention to address freely also the differential equation itself as a (generalized) gradient flow.
 
7
In the distributed-parameter variant and in the notation of Sections 4.​2.​3 and 4.​3.​4.​1, we can think of the internal variable z ≥ 0 on the contact surface \(\varGamma \mathrm{C}\) as governed by a rate-independent flow rule \({\buildrel \hspace{0.85005pt}.\over z} =\sigma _{\mathrm{n}}\vert [[{\buildrel \hspace{0.85005pt}.\over u}]]\mathrm{t}\vert /k\) (a so-called Archard’s law [24], i.e., volume of the removed debris due to wear is proportional to the work done by frictional forces) with \([[u]]\mathrm{t}\) standing for the tangential displacement and \(\sigma _{\mathrm{n}} \geq 0\) the normal stress, and k > 0 a material constant expressing resistance of the surface to wear (usually large). The interpretation of z is the width of a layer where the material was already brushed away, and as such, it enters the contact boundary condition and thus \(\mathcal{E}\). See, e.g., [20, 572].
 
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Metadaten
Titel
A general view of rate-independent systems
verfasst von
Alexander Mielke
Tomàš Roubíček
Copyright-Jahr
2015
Verlag
Springer New York
Buch
Rate-Independent Systems

Print ISBN: 978-1-4939-2705-0

Electronic ISBN: 978-1-4939-2706-7

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-1-4939-2706-7_1

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