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Emergence of rate-independent dissipation from viscous systems with wiggly energies

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Abstract

We consider the passage from viscous systems to rate-independent system in the limit of vanishing viscosity and for wiggly energies. Our new convergence approach is based on the \({(\mathcal{R},\mathcal{R}^*)}\) formulation of De Giorgi, where we pass to the Γ limit in the dissipation functional. The difficulty is that the type of dissipation changes from a quadratic functional to one that is homogeneous of degree 1, thus leading to hysteresis. The analysis uses the decomposition of the restoring force into a macroscopic part and a fluctuating part, where the latter is handled via homogenization.

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Author information

Correspondence to Alexander Mielke.

Additional information

Dedicated to Ingo Müller on the occasion of his 75th birthday.

Research partially supported by ERC-AdG no. 267802 Analysis of Multiscale Systems Driven by Functionals. The author is grateful for helpful discussions with Lev Truskinovsky and Giuseppe Savaré.

Communicated by W.H. Müller.

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Mielke, A. Emergence of rate-independent dissipation from viscous systems with wiggly energies. Continuum Mech. Thermodyn. 24, 591–606 (2012). https://doi.org/10.1007/s00161-011-0216-7

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Keywords

  • Gamma convergence for evolution
  • De Giorgi formulation
  • Rate-independent plasticity
  • Viscous gradient flow
  • Wiggly energy landscape

Mathematics Subject Classification (2010)

  • 74N30
  • 74D10
  • 74K70