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Stochastic homogenization of rate-independent systems and applications

Abstract

We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We prove some convergence results with respect to stochastic two-scale convergence, which are related to classical \(\Gamma \)-convergence results. The main result is a general \(\liminf \)-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rate-independent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandtl–Reuss plasticity, Tresca friction on a macroscopic surface and Tresca friction on microscopic fissures.

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Acknowledgements

This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through Grant CRC 1114 “Scaling Cascades in Complex Systems,” Project C05 Effective models for interfaces with many scales. The Author also thanks the reviewers for the very helpful suggestions.

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Correspondence to Martin Heida.

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Communicated by Andreas Öchsner.

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Heida, M. Stochastic homogenization of rate-independent systems and applications. Continuum Mech. Thermodyn. 29, 853–894 (2017). https://doi.org/10.1007/s00161-017-0564-z

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Keywords

  • Stochastic homogenization
  • Rate-independent
  • Hysteresis
  • Two-scale convergence
  • Plasticity
  • Tresca friction
  • Coulomb friction