Global Existence for a Three-Dimensional Model for the Thermo-MechanicalEvolution of Shape Memory Alloys

摘要

In the paper the authors study the mathematical aspects of the three-dimensionalFremond model for the thermo-mechanical phase transitions in shape memory alloys. In this model the free energy is constructed as a function of (Kelvin) temperature, macroscopic strain, strain gradient, and phase fractions. The resulting field equations are given by the balance laws of (linear) momentum and energy, coupled with an evolution variational inequality for the phase fractions. This system is highly nonlinear, and various simplified versions have been studied in the past. In this note, the authors retain a nonlinear term in the energy balance that has been neglected in former papers on mathematical reasons. The authors show the existence of a (weak) solution by introducing a suitable time discretization and then passing to the limit with the aid of a priori estimates.