Mathematics Colloquium

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Highly Irregular Microstructures and TN Configurations

Speaker: John Ball, Heriot-Watt University and Maxwell Institute for Mathematical Sciences, Edinburgh

Location: Warren Weaver Hall 1302

Date: Monday, December 8, 2025, 3:45 p.m.

Synopsis:

Remarkable martensitic microstructures are observed in the alloy Ti_{76}Nb_{22}Al_2, which undergoes a cubic to orthorhombic transformation with six martensitic variants whose nonlinear strains (positive definite symmetric matrices U_i, i=1...6) all have middle eigenvalue 1, thus allowing exact interfaces between the high temperature (cubic, austenite) phase and the low temperature (orthorhombic, martensite) phase. The unusual microstructure arises from the fact that while there are 12 matrices in the set of martensitic energy wells (the union of SO(3)U_i for i=1,...,6) which are compatible with the undistorted austenite (i.e. which are rank-one connected to the identity), these 12 matrices are pairwise incompatible. The talk will explain the underlying theory and describe attempts to understand the observed microstructures by studying gradient Young measures, exact gradients, and T_N configurations supported on these 12 incompatible matrices. This is joint work with Tomonari Inamura and Francesco Della Porta.