The production of modern multi-phase steels -- a challenge for industrial mathematics
Dietmar Hoemberg (WIAS) /
Despite the development of sophisticated composite materials in recent years, steel is still the basic construction material for industrial societies. Steel is also a modern material, e.g., 80\% of the steel grades sold by the German steel producer ThyssenKrupp have been developed in the last 15 years. In particular, this development has been triggered by the the demands of automotive industry. In 1999, a consortium of 33 international steelproducers formed the ULSAB-AVC (UltraLight Steel Auto Body-Advanced Vehicle Concepts) consortium to pursue a steel-intensive family car, fit for the 21st century, that would be safe, affordable and fuel-efficient. Besides the development of structural components such as tailored blanks and tubes the main part of the innovation came from a consequential employment of multi-phase steels. Approximately 75 \% of the ULSAB-AVC bodystructure design uses dual-phase (DP) steels. These steels have shown high potential for automotive applications due to their remarkable property combination with high strength and good formability. The standard process route for the production of DP steel is by hot rolling and subsequent controlled cooling. It provides good microstructure homogeneity with acceptable surface quality for many applications. In the first part of the talk I will review models suitable to describe the ferrite growth in DP steels. We revisit the classical Johnson-Mehl-Avrami-Kolmogorov nucleation and growth models and show how to use this approach to account also for soft impingement effects. In the second part we will ask how one can detect these phase transitions experimentally. This leads us to the consideration of dilatometers, measuring the length and temperature changes in a specimen during controlled cooling. We show how mathematics, in this case the solution of an inverse problem, can help to detect the phase transition kinetics. The macroscopic Avrami models give no information about grain size. In the third part of the talk we discuss a possibility to incorporate grain size information into our modelling approach. To this end we study a Fokker-Planck type evolution equation for the grain-size distribution. In the last part of the talk we finally consider the production of DP steel. To simulate a real run-out table (ROT) we consider the pilot hot-rolling mill at IMF Freiberg, Germany. In a first step another inverse problem has to be solved to identify the cooling conditions on ROT. Then, we set up an optimal control problem for the production of DP steel and conclude with some numerical results using a sequential quadratic programming approach and an experimental validation.