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Rate-Independent Systems pp 1–43Cite as

A general view of rate-independent systems

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

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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Notes

  1. 1.

    See., e.g., [455, Sect. 15.1.1] for this terminology.

  2. 2.

    Special situations can be handled as in [367], where the case α = 2 and \([-\lambda, 0]\) instead of \((-\infty, 0]\) is considered.

  3. 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. 4.

    Historically, internal variables have also been called “internal coordinates” or “internal degrees of freedom”; cf. [72, Ch. 6].

  5. 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. 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. 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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Authors and Affiliations

  1. Weierstraß-Institute für Angewandte Analysis und Stochastik, Berlin, Germany

    Alexander Mielke

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Alexander Mielke

  3. Mathematical Institute, Charles University, Prague, Czech Republic

    Tomàš Roubíček

  4. Institute of Thermomechanics, Czech Academy of Sciences, Prague, Czech Republic

    Tomàš Roubíček

  5. Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic

    Tomàš Roubíček

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Mielke, A., Roubíček, T. (2015). A general view of rate-independent systems. In: Rate-Independent Systems. Applied Mathematical Sciences, vol 193. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2706-7_1

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