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Existence and Uniqueness Results for a Problem of Thermoelastoplasticity with Phase Transitions
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We investigate a mathematical model of material behaviour of steel in the
context of macroscopic continuum mechanics.
Stress- and strain-dependent phase transitions (PT), transformation-induced
plasticity (TRIP) and its interactions with classical plasticity (CP) are important
phenomena which may cause distortion of workpieces. In order to simulate
real heat treatment processes of workpieces, one has to include the relevant
phenomena in a suitable mathematical model. Due to the possible interaction
of TRIP and CP the usual approach in CP without PT and TRIP has to be
modified substantially.
We obtain a system of equations resulting from the balance equations for momentum
and energy. The strain is assumed to be additively decomposed into a
thermoelastic part and a contribution from CP as well as a contribution caused
by TRIP. The PT is described by an ordinary differential equation. The strain
for TRIP is described by an ordinary differential equation as well. The evolution
of the plastic strain is given by a flow rule (cf. M. Wolff, M. Boehm, D. Helm,
Material behaviour of steel - modelling of complex phenomena and thermodynamic
consistency, Int. J. of Plast. 24, 746-774, 2008).
We prove the unique existence of a weak solution.