WIAS Preprint No. 655, (2001)

Phase-field models with hysteresis in one-dimensional thermo-visco-plasticity


  • Krejčí, Pavel
  • Sprekels, Jürgen
  • Stefanelli, Ulisse

2010 Mathematics Subject Classification

  • 34C55 35K60 47J40 74K05 74N30 80A22


  • Phase-field systems, phase transitions, hysteresis operators, thermo-visco-plasticity, thermodynamic consistency


The mathematical modelling of nonlinear thermo-visco-plastic developments and of phase transitions in solids have drawn much attention in past years. On the one hand, one is interested how phase transformations on the micro- and/or mesoscales (for instance, between different geometric configurations of the crystal lattice) influence the global thermo-visco-plastic behaviour, on the other hand, the global evolution of solid-solid phase transformations is strongly affected by the presence of micro- and/or mesoscopic stresses. In such situations, a typical macroscopic phenomenon is the occurrence of hysteresis effects, and it is therefore important to model these effects. This paper is a contribution towards this direction. A new one-dimensional model is considered that incorporates both the occurrence of hysteresis effects and of phase transitions. In this connection, the phase transition is described by the evolution of a phase-field (which is usually closely related to an order parameter of the phase transition), while the hysteresis effects are accounted for using the mathematical theory of hysteresis operators developed in the past thirty years. The model extends recent works of the first two authors on phase-field models with hysteresis to the case when mechanical effects can no longer be ignored or even prevail. It leads to a strongly nonlinear coupled system of partial differential equations in which hysteresis nonlinearities occur at several places, even under time and space derivatives. We show the thermodynamic consistency of the model, and we prove its well-posedness.

Appeared in

  • SIAM J. Math. Anal., Vol.34, 2 (2002), pp. 409-434

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