Averaging of time-periodic dissipation potentials in rate-independent processes

Martin Heida; Alexander Mielke

doi:10.3934/dcdss.2017070

Article Contents

2017, Volume 10, Issue 6: 1303-1327. Doi: 10.3934/dcdss.2017070

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Averaging of time-periodic dissipation potentials in rate-independent processes

Martin Heida^1, and
Alexander Mielke^1,2,

1.
Weierstraẞ-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39,10117 Berlin, Germany
2.
Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25,12489 Berlin, Germany

Received: October 31, 2016

Revised: December 31, 2016

Published: December 2017

M.H. was financed by Deutsche Forschungsgemeinschaft (DFG) through Grant CRC 1114 Scaling Cascades in Complex Systems, Project C05 Effective models for interfaces with many scales.
A.M. was partially supported by ERC through AdG 267802 AnaMultiScale

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Abstract

We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period $ \varepsilon$. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit $ \varepsilon \to 0$ and show that the effective dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit $ \varepsilon \to 0$.

Keywords:

Rate-independent systems,
play operator with time-dependent thresholds,
energetic solutions,
locomotion,
sweeping process.

Mathematics Subject Classification: Primary: 34C55, 47J20, 49J40, 74N30.

Citation:

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Figure 1.1. Because of the in-built unbalance, the plate compactor vibrates vertically leading to an oscillatory normal pressure. When pushing the plate compactor horizontally it will move only when the normal pressure is very low

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Figure 1.2. (A) In rest, the woodpecker sticks to the metal rod by dry friction, when oscillating the reduction in friction allows for a slow sliding downwards, cf. [20]. (B) Toy ramp walker: the frog walks down only, when alternating the weight between the rigid downhill leg and the hinged uphill leg. (C) Rocking animal: A weight beyond the table edge pulls the cow forward, while the perpendicular rocking motions allows the lifted legs to swing forward because of the reduced normal pressure

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Figure 2.1. The bold, red curve is the solution of $0\in \rho(t/\varepsilon)\mathrm{Sign}(\dot y(t)) + y(t) -5t +t^2$ with $y(0)=0$ for $\varepsilon =0.04$. The shaded, wavy area indicates the stable regions

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Figure 2.2. Plots for the solution of (2.2). (A) The positions $y_j(t)$ of the two legs move by alternating between plateaus (sticking phase) and fast motion. (B) The derivatives $\dot y_j(t)$ show that the motion is alternating, i.e. at most one of the legs moves at a time. (C) The path $t\mapsto y(t)=(y_1(t),y_2(t)) \in \mathbb{R}^2$ shows a microscopic zigzag pattern.

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