Lectures in the Winter term 2012 / 2013
Topics in Applied Analysis: Microstructure in Shape Memory Alloys
- Date: Tuesday, 9:00 - 11:00 h
- in case the lecture cannot take place due to travel etc.:
- Thursday, 9:00 - 11:00 h
- Location: MPI MiS, room A01
- First lecture: October 9th
We start with an introduction into elasticity theory, in particular the notion of strain (in the geometrically linear setting). We then introduce the notion of shape memory alloys, with their transition from a high symmetry Austenite phase to a low symmetry Martensite phase; this will involve a bit of symmetry considerations (point group etc).
We then will address Martensitic twins and their compatibility with Austenite, establishing "rigidity" for certain symmetries (like square-to-rectangular in 2-d and cubic-to-tetragonal in 3-d) and non-rigidity in other cases (like hexagonal-to-rhombic in 2-d). The non-rigidity results will be obtained via the tool of "convex integration". We then will introduce interfacial energy as a means to select physically relevent microstructure. In the rigid cases, the optimal microstructure is a branched Martensitic twin. This involves establishing matching upper and lower bounds for the minimal energy (that is a sum of elastic and interfacial energy); obtaining the lower bounds involves suitable interpolation inequalities. We also plan to address nucleation barriers for Martensite in Austenite in that context.