The current work contains experimental and theoretical work about the formation of martensite from the austenitic state of the steel Hardox 450. Simulation of rolling and subsequent quenching of martensitic steel plates requires a model that can account for previous deformation, current stresses and the temperature history, therefore dilatometry experiments were performed, with and without deformation. Two austenitization schedules were used and in the standard dilatometry the cooling rates varied between 5-100 °C/s, in order to find the minimum cooling rate that gives a fully martensitic microstructure. Cooling rates larger than 40°C/s gave a fully martensitic microstructure. The cooling rate of 100 °C/s was used in the deformation dilatometry tests where the uniaxial deformation varied from 5-50 %. The theoretical work involved modelling of the martensite formation and the thermal/transformation strains they cause in the steel. Characterizations were done using light optical microscopy, hardness tests and electron backscatter diffraction technique. The parent austenite grains of the martensitic structure were reconstructed using the orientation relationship between the parent austenite and the martensite. Kurdjumov-Sachs orientation relationships have previously been proven to work well for low-carbon steels and was therefore selected. The standard implementation of the Koistinen-Marburger equation for martensite formation and a more convenient approach were compared. The latter approach does not require the storage of initial austenite fraction at start of martensite formation. The comparison shows that the latter model works equally well for the martensite formation. The results showed that the use of martensite start and finish temperatures calibrated versus experiments for Hardox 450 works better when computing thermal expansion than use of general relations based on the chemistry of the steel. The results from deformation dilatometry showed that deformation by compressive uniaxial stresses impedes the martensite transformation. The simplified incremental model works well for deformation with 5 % and 10 %. However, the waviness in the experimental curve for deformation 50 % does not fit the model due vi to large barrelling effect and the large relative expansion for the material that the sample holders are made of. Crystallographic reconstruction of parent austenite grains were performed on a hot-rolled as-received reference sample and dilatometry samples cooled with 60 °C/s and 100 °C/s. The misorientation results showed that the samples match with the Kurdjumov-Sachs orientation relationship in both hot rolled product and dilatometry samples. When misorientation between adjacent pixels are between 15° and 48°, then the boundary between them was considered as a parent austenite grain. The austenitic grain boundaries of the sample cooled at 100 °C/s is in general identical with the hot rolled sample when considering high angle boundaries (15°-48°). The results from the hardness tests showed that the rolled product exhibits higher hardness as compared to samples cooled by 100 °C/s and 60 °C/s. This can be attributed to the formation of transition-iron-carbides in the hot rolled product due to longer exposure of coiling temperature.

The current work contains experimental and theoretical work about the formation of martensite from the austenitic state of the steel Hardox 450. Simulation of rolling and subsequent quenching of martensitic steel plates requires a model that can account for previous deformation, current stresses and the temperature history, therefore dilatometry experiments were performed, with and without deformation. Two austenitization schedules were used and in the standard dilatometry the cooling rates varied between 5-100 °C/s, in order to find the minimum cooling rate that gives a fully martensitic microstructure. Cooling rates larger than 40°C/s gave a fully martensitic microstructure. The cooling rate of 100 °C/s was used in the deformation dilatometry tests where the uniaxial deformation varied from 5-50 %. The theoretical work involved modelling of the martensite formation and the thermal/transformation strains they cause in the steel. Characterizations were done using light optical microscopy, hardness tests and electron backscatter diffraction technique. The parent austenite grains of the martensitic structure were reconstructed using the orientation relationship between the parent austenite and the martensite. Kurdjumov-Sachs orientation relationships have previously been proven to work well for low-carbon steels and was therefore selected. The standard implementation of the Koistinen-Marburger equation for martensite formation and a more convenient approach were compared. The latter approach does not require the storage of initial austenite fraction at start of martensite formation. The comparison shows that the latter model works equally well for the martensite formation. The results showed that the use of martensite start and finish temperatures calibrated versus experiments for Hardox 450 works better when computing thermal expansion than use of general relations based on the chemistry of the steel. The results from deformation dilatometry showed that deformation by compressive uniaxial stresses impedes the martensite transformation. The simplified incremental model works well for deformation with 5 % and 10 %. However, the waviness in the experimental curve for deformation 50 % does not fit the model due vi to large barrelling effect and the large relative expansion for the material that the sample holders are made of. Crystallographic reconstruction of parent austenite grains were performed on a hot-rolled as-received reference sample and dilatometry samples cooled with 60 °C/s and 100 °C/s. The misorientation results showed that the samples match with the Kurdjumov-Sachs orientation relationship in both hot rolled product and dilatometry samples. When misorientation between adjacent pixels are between 15° and 48°, then the boundary between them was considered as a parent austenite grain. The austenitic grain boundaries of the sample cooled at 100 °C/s is in general identical with the hot rolled sample when considering high angle boundaries (15°-48°). The results from the hardness tests showed that the rolled product exhibits higher hardness as compared to samples cooled by 100 °C/s and 60 °C/s. This can be attributed to the formation of transition-iron-carbides in the hot rolled product due to longer exposure of coiling temperature.

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.