Variational models in martensitic phase transformations with applications to steels

Thesis

This thesis concerns the mathematical modelling of phase transformations with a special emphasis on martensitic phase transformations and their application to the modelling of steels.

In Chapter 1, we develop a framework that determines the optimal transformation strain between any two Bravais lattices and use it to give a rigorous proof of a conjecture by E.C. Bain in 1924 on the optimality of the so-called Bain strain.

In Chapter 2, we review the Ball-James model and related concepts. We present some simplification of existing results.

In Chapter 3, we pose a conjecture for the explicit form of the quasiconvex hull of the three tetragonal wells, known as the three-well problem. We present a new approach to finding inner and outer bounds.

In Chapter 4, we focus on highly compatible, so called self-accommodating, martensitic structures and present new results on their fine properties such as estimates on their minimum complexity and bounds on the relative proportion of each martensitic variant in them.

In Chapter 5, we investigate the contrary situation when self-accommodating microstructures do not exist. We determine, whether in this situation, it is still energetically favourable to nucleate martensite within austenite. By constructing different types of inclusions, we find that the optimal shape of an inclusion is flat and thin which is in agreement with experimental observation.

In Chapter 6, we introduce a mechanism that identifies transformation strains with orientation relationships. This mechanism allows us to develop a simpler, strain-based approach to phase transformation models in steels. One novelty of this approach is the derivation of an explicit dependence of the orientation relationships on the ratio of tetragonality of the product phase.

In Chapter 7, we establish a correspondence between common phenomenological models for steels and the Ball-James model. This correspondence is then used to develop a new theory for the (5 5 7) lath transformation in low-carbon steels. Compared to existing theories, this new approach requires a significantly smaller number of input parameters. Furthermore, it predicts a microstructure morphology which differs from what is conventionally believed.

Authors: Muehlemann, A

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: austenite martensite interfaces; double shear theories; austenite; Terephthalic Acid; transformation strain; (557) habit planes; Nishiyama-Wassermann; lath martensite; non-linear elasticity; Bravais lattices; twins within twins; Pitsch; tetragonal; microstructure; Relaxation; lattice transformation; least atomic movement; Quasiconvexity; non-classical interfaces; Greninger-Troiano; volume fractions; orientation relationships; fcc-to-bcc; self-accommodation; steel; twinned martensite; Kurdjumov-Sachs; Bain strain; Laminates; martensite
• Subjects: Calculus of variations; Phase Transformation; Energy Minimisation; martensite