Penrose-Fife model with activated phase transformation - Existence and effective model for slow-loading regimes.

Abstract: We study a non-autonomous system of two evolutionary PDEs, an enhanced version of the Penrose-Fife model introduced in 1990, which first in history coupled the Allen-Cahn phase-field equation with thermal effects in a thermodynamically consistent way. In our work we consider an activated phase transformation which makes the phase-field equation non-linear in time derivative; it contains an additional term that describes also the Coulomb dry friction and whose analogues appear in rate-independent models of plasticity or damage. Apart from proving the existence of solutions of the system, we investigate its effective behavior for the so-called slow-loading regimes. There the internal time-scale introduced by phase-field viscosity and thermal conductivity is much smaller than the time-scale of the loading, an external force acting on the system. Hence the system stays almost in its equilibrium described by the static solution. However, the limit of solutions (as the ratio of the time scales tends to zero) may not be characterized by the corresponding rate-independent model where the viscous-like terms have been dropped. The reason is that the phase transformation is driven by a non-convex thermodynamic potential and hence the solutions may develop jumps in the limit. In order to resolve the jump's detailed trajectory, along which the viscous effects play a crucial role, we use the so-called Balanced-Viscosity (BV) solutions introduced by Efendiev and Mielke in 2006. This work is a collaboration with Matthias Liero and Alexander Mielke from WIAS Berlin.

Penrose O., Fife P. C. - Thermodynamically Consistent Models of Phase-Field Type for the Kinetics of Phase Transitions (1990)

Sprekels J., Zheng S. M. - Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions (1993)

Efendiev M., Mielke A. - On the rate-independent limit of systems with dry friction and small viscosity (2006)

Mielke A., Roubicek T. - Rate-independent Systems Theory and Application (2015)

MathematicsPhysics

Audience: researchers in the topic

---

Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor).

---