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Models for self-similarity and disclinations in martensite

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DNM - The mathematical design of new materials

The austenite-to-martensite phase-transformation is a first-order diffusionless transition occurring in elastic
crystals and characterized by an abrupt change of shape of the underlying crystal lattice. It manifests itself
to what in materials science is called a martensitic microstructure, an intricate highly inhomogeneous
pattern populated by sharp interfaces that separate thin plates composed of mixtures of different martensitic
phases (i.e., rotated copies of a low symmetry lattice) possibly rich in defects and lattice mismatches. In
this talk we review a series of separate results on the modeling of inter-connected phenomena observed in
martensite, which are self-similarity (criticality) and disclinations.
Inspired by Bak’s cellular automaton model for sand piles, we introduce a conceptual model for a
martensitic phase transition and analyze the properties of the patterns obtained. Nucleation and evolution
of martensitic variants is modeled as a fragmentation process in which the microstructure evolves via
formation of thin plates of martensite embedded in a medium representing the austenite. While the
orientation and direction of propagation of the interfaces separating the plates is determined by kinematic
compatibility of the crystal phases, their nucleation sites are inevitably influenced by defects and disorder,
which are encoded in the model by means of random variables. We investigate distribution of the lengths
of the interfaces in the pattern and establish limit theorems for some of the asymptotics of the interface
profile. We also discuss numerical aspects of determining the behavior of the density profile and power
laws from simulations of the model and present comparisons with experimental data.
Turning our attention on defects, we investigate wedge disclinations, high-energy rotational defects caused
by an angular lattice mismatch that were predicted by Volterra in his celebrated 1907 paper. Unlike
dislocations, which have received considerable attention since the 1930s, disclinations have received
disproportionally less interest. However, disclinations are not uncommon as they accompany, as a relevant
example, rotated and nested interfaces separating (almost) kinematically compatible variants as in
martensitic avalanche experiments. Here we follow two modeling approaches. First, we introduce a few
recent results on the modeling of planar wedge disclinations in a continuum, purely (non-linear) elastic
model that describes disclinations as solutions of some differential inclusion. Secondly an atomistic model
of nearest-neighbor interactions over a triangular lattice inspired by the literature on discrete models for
Some of these results are from a collaboration with J.M. Ball and B. Hambly (Oxford) and P. Van Meurs

This talk is part of the Isaac Newton Institute Seminar Series series.

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