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Analysis Seminar

Thursday, January 25, 2018 - 12:00pm

Robert V. Kohn

New York University


University of Pennsylvania


Please note that this week's Analysis Seminar is at a special time of 12:00 pm, not the usual 3:00 pm. Also, we are in a different room from usual - 4E9 as opposed to the usual 4C8.

In martensitic phase transformation, the formation of microstructure is due to minimization of elastic plus surface energy. There is by now a lot of work on macroscopically-uniform microstructures. In the presence of stress or a thermal gradient, however, one expects microstructures whose volume fractions and other features vary macroscopically. I'll discuss some model problems of this kind, involving non-uniform twinning of two martensite variants. In these problems the loads or boundary conditions require the volume fractions to vary; this forces the twin boundaries to tilt away from their stress-free orientations, and leads to what might be called a "zig-zag microstructure." The goals for mathematical analysis are (a) to show that the zig-zag microstructure achieves an optimal energy scaling law, (b) to explore the local properties of energy-minimizing patterns, and (c) to understand the robust periodicity observed in experimental realizations of such microstructures. I'll discuss some recent progress toward (a) and (b) (joint work with Alex Misiats and Stefan Mueller). The mathematical models will be described from scratch (no prior knowledge will be assumed concerning martensitic phase transformation).