Selection mechanisms for microstructures and reversible martensitic transformations
Keywords: Shape memory alloys
Partial Differential Equation
Reversibility
Martensitic transformations
Calculus of Variations
Nonlinear analysis

The work in this thesis is inspired by the fabrication of Zn45Au30Cu25. This is the first alloy undergoing ultra-reversible martensitic transformations and closely satisfying the cofactor conditions, particular conditions of geometric compatibility between phases, which were conjectured to influence reversibility. With the aim of better understanding reversibility, in this thesis we study the martensitic microstructures arising during thermal cycling in Zn45Au30Cu25, which are complex and different in every phase transformation cycle.
Our study is developed in the context of continuum mechanics and nonlinear elasticity, and we use tools from nonlinear analysis.
The first aim of this thesis is to advance our understanding of conditions of geometric compatibility between phases. To this end, first, we further investigate cofactor conditions and introduce a physically-based metric to measure how closely these are satisfied in real materials. Secondly, we introduce further conditions of compatibility and show that these are nearly satisfied by some twins in Zn45Au30Cu25. These might influence reversibility as they improve compatibility between high and low temperature phases.
Martensitic phase transitions in Zn45Au30Cu25 are a complex phenomenon, especially because the crystalline structure of the material changes from a cubic to a monoclinic symmetry, and hence the energy of the system has twelve wells. There exist infinitely many energy-minimising microstructures, limiting our understanding of the phenomenon as well as our ability to predict it. Therefore, the second aim of this thesis is to find criteria to select physically-relevant energy minimisers. We introduce two criteria or selection mechanisms. The first involves a moving mask approximation, which allows one to describe some experimental observations on the dynamics, while the second is based on using vanishing interface energy.
The moving mask approximation reflects the idea of a moving curtain covering and uncovering microstructures during the phase transition, as appears to be the case for Zn45Au30Cu25, and many other materials during thermally induced transformations. We show that the moving mask approximation can be framed in the context of a model for the dynamics of nonlinear elastic bodies. We prove that every macroscopic deformation gradient satisfying the moving mask approximation must be of the form
1 + a(x) ⊗ n(x), for a.e. x.
With regards to vanishing interface energy, we consider a one-dimensional energy functional with three wells, which simplifies the physically relevant model for martensitic transformations, but at the same time highlights some key issues. Our energy functional admits infinitely many minimising gradient Young measures, representing energy-minimising microstructures. In order to select the physically relevant ones, we show that minimisers of a regularised energy, where the second derivatives are penalised, generate a unique minimising gradient Young measure as the perturbation vanishes.
The results developed in this thesis are motivated by the study of Zn45Au30Cu25, but their relevance is not limited to this material. The results on the cofactor conditions developed here can help for the understanding of new alloys undergoing ultra-reversible transformations, and as a guideline for the fabrication of future materials. Furthermore, the selection mechanisms studied in this work can be useful in selecting physically relevant microstructures not only in Zn45Au30Cu25, but also in other materials undergoing martensitic transformations, and other phenomena where pattern formation is observed.