Metastable austenitic and duplex stainless steels are widely used materials in industrial anddomestic applications, owing to their attractive characteristics such as good corrosion resistanceand favorable mechanical properties. Both types of steel experience enhanced mechanicalproperties during plastic deformation due to the formation of the martensite phase from theparent austenite phase, this is called deformation-induced martensitic transformation (DIMT).It is therefore of technical interest to study the transformation mechanism and its impact onmechanical properties for a better understanding and ultimately for developing new materialswith improved performance in certain applications. In the present thesis, two austenitic stainless steels (201Cu, HyTens® 301) and two duplexstainless steels (FDX25®, FDX27®) were investigated. Samples were tensile tested during insitusynchrotron radiation experiments performed at the Cornell High Energy SynchrotronSource (CHESS), Ithaca, USA. Tests were performed at both room temperature and at elevatedtemperatures. The collected diffraction data were then processed by software such as Fit2D andMATLAB. Quantitative phase fraction analysis based on the direct comparison method wasperformed successfully. Microstructural analysis of samples before deformation and after thefull tensile testing was also performed using electron microscopy. The deformation induced martensitic transformation took place in HyTens 301, FDX25 andFDX27, but in 201Cu the austenite was stable during the tensile tests conducted here. The a’-martensite formed in a significantly higher fraction than the ε-martensite in all alloys. At roomtemperature, the critical stress levels for martensitic transformation were 490 MPa, 700 MPaand 700MPa for HyTens 301, FDX25 and FDX27, respectively.

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.