Archive for Rational Mechanics and Analysis

, Volume 162, Issue 2, pp 137–177 | Cite as

A Variational Formulation of¶Rate-Independent Phase Transformations¶Using an Extremum Principle

  • Alexander Mielke
  • Florian Theil
  • Valery I. Levitas
Article

Abstract

We propose a rate-independent, mesoscopic model for the hysteretic evolution of phase transformations in shape-memory alloys. The model uses the deformation and phase-indicator function as basic unknowns and the potentials for the elastic energy and for the dissipation as constitutive laws. Using the associated functionals, admissible processes are defined to be the ones which are stable at all times and which satisfy the energy inequality.

This concept leads to a natural time-incremental method which consists in a minimization problem. The mesoscopic model is obtained by a relaxation procedure. It leads to new functionals involving the cross-quasiconvexification of the elastic stored-energy density. For a special case involving two phases of linearized elastic materials we show that the incremental problem provides existence of admissible processes for the time-continuous problem, if we let the time-step go to 0.

Dedicated to Erwin Stein on the occasion of his seventieth birthday

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Alexander Mielke
    • 1
  • Florian Theil
    • 2
  • Valery I. Levitas
    • 3
  1. 1.Mathematisches Institut A¶Universität Stuttgart¶Pfaffenwaldring 57¶70569 Stuttgart, Germany¶e-mail: mielke@mathematik.uni-stuttgart.deDE
  2. 2.University of Warwick¶Mathematics Institute¶CV4 7AL Coventry, UKGB
  3. 3.Texas Tech University¶Department of Mechanical Engineering¶Lubbock, TX 79409-1021, USAUS

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