Steel is an important material in various applications and has a complex material behaviour which is - in addition to elastic and plastic effects - basically characterized by phase transitions. An important feature in this context is the transformation-induced plasticity, which arises during phase transitions and leads to permanent deformation at deviatoric stress (below the yield point). Moreover, the material properties of steel often depend on the carbon content which makes the whole situation even more complex, in particular if an inhomogeneous carbon distribution is considered in the context of special heat treatment processes like the so-called carburization. Modelling of the main phenomena of the material behaviour of steel leads to a system of coupled nonlinear partial and ordinary differential equations for the time- and space-dependent temperature, displacement and phase fractions, coupled with a variational inequality which takes classical plasticity into account. A mathematical investigation of this initial-boundary value problem is not only of independent mathematical interest, but also of great importance for the numerical simulation of the material behaviour of steel workpieces in certain highly relevant situations, such as quenching or case hardening.

The main objective is to prove existence and uniqueness of a solution of the corresponding initial-boundary value problem for the coupled problem of linear thermo-elasto-plasticity including phase transitions and transformation-induced plasticity under appropriate conditions. In close cooperation with the AG Numerik PDE some preparations for the numerical implementation are under way. The work will be accompanied by an extensive workpiece simulation for relevant cases with COMSOL™ or MATLAB™, where the measured data is provided by the Foundation Institute for Materials Science (IWT) Bremen. Moreover, a complex implementation (3D calculations with real data and comparison with commercial software packages) could follow up this work as a larger project. In addition the investigation of special heat treatment processes such as carburizing, nitriding or nitrocarburizing take a great interest.