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Balanced viscosity solutions to a rate-independent system for damage

Published online by Cambridge University Press:  24 January 2018

DOROTHEE KNEES ,
RICCARDA ROSSI  and
CHIARA ZANINI
DOROTHEE KNEES
Affiliation:
Institute of Mathematics, University of Kassel, Heinrich-Plett Str. 40, 34132 Kassel, Germany email: dknees@uni-kassel.de
RICCARDA ROSSI
Affiliation:
Department DIMI, University of Brescia, Via Branze 38, 25133 Brescia, Italy email: riccarda.rossi@unibs.it
CHIARA ZANINI
Affiliation:
Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy email: chiara.zanini@polito.it
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Abstract

This article is the third one in a series of papers by the authors on vanishing-viscosity solutions to rate-independent damage systems. While in the first two papers (Knees, D. et al. 2013 Math. Models Methods Appl. Sci.23(4), 565–616; Knees, D. et al. 2015 Nonlinear Anal. Real World Appl.24, 126–162) the assumptions on the spatial domain Ω were kept as general as possible (i.e., non-smooth domain with mixed boundary conditions), we assume here that ∂Ω is smooth and that the type of boundary conditions does not change. This smoother setting allows us to derive enhanced regularity spatial properties both for the displacement and damage fields. Thus, we are in a position to work with a stronger solution notion at the level of the viscous approximating system. The vanishing-viscosity analysis then leads us to obtain the existence of a stronger solution concept for the rate-independent limit system. Furthermore, in comparison to [18, 19], in our vanishing-viscosity analysis we do not switch to an artificial arc-length parameterization of the trajectories but we stay with the true physical time. The resulting concept of Balanced Viscosity solution to the rate-independent damage system thus encodes a more explicit characterization of the system behaviour at time discontinuities of the solution.

Keywords

Rate-independent damage systemvanishing-viscosity approximationBalanced Viscosity solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 1 , February 2019 , pp. 117 - 175
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

D. Knees acknowledges the partial financial support through the DFG-Priority Program SPP 1962 Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization.

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