Project Details
Projekt
Regularity and Scaling of Wild Microstructures in the Modelling of Shape-Memory Alloys
Applicant
Professorin Dr. Angkana Rüland
Subject Area
Mathematics
Term
since 2020
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 441068247
The modelling of shape-memory alloys provides a prototypical, highly non-convex problem originating from materials science in which a mathematically striking dichotomy between rigidity and flexibility occurs: On the one hand, mathematical models which are complemented with comparably high regularity conditions are typically rigid, in the sense that the solutions to the associated differential inclusions follow kinematic compatibility conditions. Experimentally this is observed, for instance, in the presence of rather regular, so-called twin or crossing twin microstructures. On the other hand, for many mathematical models at low regularity a plethora of highly non-unique, typically very complex and irregular, so-called ‘wild’ solutions emerge. In the first funding phase of the SPP we made important progress towards the understanding of this dichotomy by, for instance, deriving sharp scaling results for the Tartar square, i.e. by providing the first scaling results in regimes in which a (weak) dichotomy between rigidity and flexibility is present. Moreover, combining perspectives from the calculus of variations and harmonic analysis and building on the analysis of the Tartar square, we developed a versatile, robust tool kit which also allowed us to address (sharp) estimates for the scaling of nucleation barriers in problems involving a higher degree of flexibility (in the form of higher order laminates). In this project we seek to proceed beyond the regime of the weak dichotomy between rigidity and flexibility by addressing scaling properties of carefully selected model problems displaying the strong dichotomy between rigidity and flexibility. Moreover, it is our objective to study selected physical phase transformations with gauges (in the form of geometrically linear or nonlinear frame indifference), eventually allowing for predictions of length scale distributions in the resulting microstructures and in experiments. Finally, we plan to explore the fine properties of minimizers and minimizing sequences in scaling laws with higher order laminates, capturing their more detailed properties. This will eventually allow for comparisons of these theoretical predictions with experimental observations of complex microstructures.
DFG Programme
Priority Programmes
