In this dissertation the formation of precipitates in steel is studied.
In the first chapter some theoretical background is presented to the topic.
Then in the second chapter an analytical approximation to the classical nucleation and growth model is presented.
In the third chapter a reference-free modified embedded atom method (RF-MEAM) potential is constructed for Fe, suitable for ferrite and austenite.
In chapter four the formation of Nb-C and Ti-C clusters in ferrite is studied, here formation paths with a monotone decrease in Gibbs energy are found. Moreover, the resulting clusters remain stable up to 1100K.
In the fifth chapter other group IV and V transition metal-carbon systems in ferrite are studied in small cells using density functional theory. Showing similar energy patterns as previously found for Nb and Ti. Furthermore the transformation from a metal-carbon cluster to a carbide precipitate is studied, where the matrix strain is found to be the driving force behind the transformation.
in the sixth and last chapter some recommendations are presented.

Advanced high strength steels (AHSS) have been widely employed in the automotive industry to meet the requirements of improving crash performance while reducing vehicle weight. The excellent performance of AHSS in strength and ductility comes from a dedicated design of alloy composition and microstructure. However, the addition of some alloying elements may lead to poor weldability. Solidification cracking tends to occur in AHSS containing detrimental elements such as phosphorous or sulfur under unfavorable welding conditions.

Solidification cracking is a complex problem associated with multiple physical phenomena at different length scales. During welding, with a localized heat input, the material is heated to melt, forming a weld pool. Upon cooling, a mushy zone consisting of solid grains and liquid channels forms behind the weld pool. As temperature drops, solid grains in the mushy zone grow, which tends to close liquid channels in the mushy zone. However, during solidification, detrimental elements segregate in the liquid, inhibiting the closure of the liquid channel. Meanwhile, due to solidification shrinkage and thermal contraction, the liquid channels experience a tensile load, which tends to open the liquid channel. If there is insufficient liquid feeding, solidification cracking occurs. Solidification cracking can be avoided by controlling parameters including alloy composition, microstructure and processing conditions like power, welding velocity, laser beam shapes, etc.

Due to the complexity, an accurate prediction of solidification cracking under various welding conditions is challenging. A full-field simulation of a welded component which incorporates all the physical phenomena and is capable of handling various welding conditions is not realistic with current computational power. To achieve an accurate prediction of solidification cracking, approximations and simplifications must be made while major physical mechanisms are still properly captured. Moreover, existing modelling techniques should be improved or adapted to fulfill the requirements of welding simulations.

This research starts from explicitly modelling segregation and solidification during welding. In the literature, cellular automata (CA) models and phase field (PF) models have been widely used to simulate segregation and solidified microstructure. The CA method is more computationally efficient compared to the PF method and thus is mainly adopted in the current research. A cellular automata solidification model is developed, where the growth velocity is determined based on kinetic undercooling at the interface. The state-of-the-art model incorporates a decentered growth algorithm to suppress grid anisotropy and a generalized height function method to calculate curvature accurately. To remove the dependency on the mesh size, a new diffusion term is proposed to handle the diffusion between the interface cells and liquid cells. Moreover, a solute redistribution method has been applied for each interface cell to resolve the mass balance error introduced by the virtual liquid cell assumption. The developed CA model is validated by simulating single-dendritic solidification in an Al-3Cu (wt.\%) alloy. The simulated tip velocities agree with the prediction of the Kurz-Giovanola-Trivedi (KGT) model. With improvements in the aspects of mesh-size independency and mass balance, the developed CA model is suitable for solidification simulation with a high undercooling, as is common in welding. It also provides an easier way to achieve multi-component solidification simulation compared to conventional CA solidification models, which need to solve a system of mass balance equations in interface cells. Despite the improvements made, due to the poor discretization of the solid-liquid interface, the CA method is less accurate compared to the phase field method. Up to now, no CA models can reproduce the dendrite tip velocity predicted by a Green function method in single-dendritic solidification simulations.

For better accuracy, the research was extended with a focus on PF modelling of segregation in the liquid channels in the mushy zone during welding. Following the frozen temperature gradient approximation, the complex thermal conditions during welding are approximated with a directional solidification condition defined by a constant temperature gradient and pulling velocity, which is the moving velocity of the liquidus isotherm. Under directional solidification conditions, columnar dendrite grains form and liquid channels exist in between neighboring grains, where solutes accumulate. The liquid channel structure and segregation is simulated with the PF model, while the solidification cracking susceptibility (SCS) is then quantified by calculating the pressure drop from the dendrite tip to the coalescence point of the liquid channels with an analytical model, the Rappaz-Drezet-Gremaud (RDG) model. A larger pressure drop represents a larger SCS. With the modelling setup, the influence of the temperature gradient and the pulling velocity on SCS have been investigated. Increasing the pulling velocity or decreasing the temperature gradient increases the pressure drop from the dendrite tip to the coalescence point, leading to an increase in SCS. Decreasing the primary dendrite arm spacing (PDAS) decreases the permeability of the liquid channel and the liquid channel length at the same time, resulting in a decrease in the pressure drop and SCS when the PDAS is small. Consideration of the PDAS dependency on the temperature gradient and the pulling velocity influences the value of the pressure drop but does not change the tendency of the SCS. The findings indicate that solidification cracking can be avoided by either decreasing the pulling velocity or increasing the temperature gradient or refining the grain size, as supported by experimental results. However, due to the high computational cost, the size of the simulation domain is limited. Moreover, the influence of process parameters on SCS cannot be considered directly due to the lack of macroscopic modelling.

To include the influence of process parameters, macroscopic thermal-mechanical modelling and microstructure modelling of the whole weld pool are necessary. The former is relatively easy to achieve, while the latter is, nevertheless, not achievable with the aforementioned CA model or PF model, as both models require a fine mesh size to simulate the liquid channel structure and segregation within the liquid channels. Simulating the solidification microstructure for the whole weld pool or the whole mushy zone requires a huge amount of computational resources. As an alternative to the numerical solution, the segregation in the liquid channel can be calculated analytically with the Scheil-Gulliver calculation, while the remaining question is how to achieve the microstructure simulation of the whole weld pool.

Therefore, the research addressed microstructure modelling of the whole weld pool. To this purpose, a special kind of CA model is adopted, which calculates the growth velocity as a function of local undercooling based on analytical models (LGK model or KGT model). In this case, there is no need to solve the concentration profiles numerically, which permits a coarse mesh size and thus a reduction in the requirement of computational resources. These kind of CA models are called CAFE models, as such CA models are always coupled with a thermal finite element (FE) model. In this research, a CAFE model which is two orders of magnitude faster than conventional CAFE models has been developed. The acceleration comes from three different sources. Firstly, by adopting an exact temporal integration and a multi-layer capture algorithm, a large time step can be employed without compromising the simulation results. Secondly, the parallelism of the simulation codes is achieved in a shared-memory environment, enabling a more efficient load balance. Thirdly, a subdomain activation and deactivation method is employed to reduce the computation tasks. The proposed model is validated by simulating the grain morphology and texture of additively manufactured samples. A good agreement is achieved between the simulations and the experiments.

By coupling the CAFE model with a thermo-mechanical FE model and a granular model to calculate liquid pressure, a multi-physics multi-scale modelling framework is developed to predict solidification cracking. The thermo-mechanical FE model calculates the profiles of temperature and strain rate for the welded component during welding; the CAFE model simulates the solidification microstructure in the whole weld pool; and the granular model calculates the pressure drop in the liquid channel network determined based on the simulated microstructure and the Scheil-Gulliver calculations. The developed modelling framework is then validated by simulating welding experiments of a TRIP steel. With a constant ratio between the power and the welding velocity, increasing the welding velocity increases the maximum pressure drop in the mushy zone, indicating an increase in SCS. Grain refinement or decreasing the freezing temperature by changing the alloy composition leads to a decrease in the maximum pressure drop in the mushy zone, representing a decrease in SCS. The predictions from the modelling framework are supported by experimental findings in the literature.

In conclusions, the research starts from microstructure modelling of segregation and liquid channel structure with both the CA method and the PF method and finalizes with a multi-physics multi-scale modelling framework of solidification cracking. In this approach, several approximations and simplifications have been made to reach the final modelling framework for solidification cracking. The developed multi-scale multi-physics modelling framework incorporates major physical mechanisms at different length scales and paves a way to understand and predict solidification cracking under various welding conditions. It provides a theoretical basis to eliminate solidification cracking by tuning parameters including alloy composition, microstructure and processing parameters like power, welding velocity and laser beam shapes, etc. Moreover, with the acceleration in microstructure simulation, the developed CAFE model contributes to the development of a digital twin of additive manufacturing.

Dual-phase (DP) steels are an important class of advanced high-strength steels (AHSS) and constitute a major share of steels for the automotive industry. A microstructure consisting of hard martensite embedded in a soft ferritic matrix gives them a good combination of strength and ductility. The martensite formation in the microstructure from austenite involves a shape and volume change, which is accommodated by the deformation of the surrounding ferritic matrix. This accommodation is known to impart typical characteristics in DP steels such as the absence of a yield point, continuous yielding and high initial work hardening rate. This thesis is an attempt to understand and model the aforementioned accommodation process in the ferritic matrix of DP steels. Traditionally, in predictive modelling of DP steel mechanical behaviour, the region of ferrite which undergoes deformation to accommodate martensitic transformation is taken into consideration as a constant thin layer of strain-hardened ferrite at the ferrite/martensite interface. This approach is shown to be inadequate for capturing local variations in ferrite deformation. Hence, electron backscatter diffraction (EBSD) experiments were carried out to study in detail the influence of various microstructural features on local variations in the transformation-induced deformation of ferrite. It was found that the crystallographic orientation of ferrite grains, martensite variant and its prior austenite grain (PAG) play an important role in determining the extent of transformation-induced deformation of ferrite. Taking a cue from this, a novel methodology comprising sequential experimental and numerical research on DP steels is developed which combines the results of PAG reconstruction, phenomenological theory of martensite crystallography (PTMC) and EBSD orientation data to estimate ferrite deformation due to every martensitic variant formed, via full-field micromechanical calculations on a virtual DP steel microstructure. Furthermore, the influence of self-accommodation during martensite variant formation on transformation-induced deformation of ferrite was also investigated. It is shown that the higher the number of variants which form from a PAG, the less the deformation caused by that PAG in the surrounding ferritic matrix. This is because of a decrease in the effective magnitude of the shear component of martensitic transformation during multi-variant transformation. The scientific findings presented in this work can be used for developing predictive models for the mechanical behaviour of not only DP steels but any multiphase steels which exhibit plastic accommodation and residual stresses in their microstructure due to martensitic phase transformation.

Inelastic deformation is a common but often neglected phenomenon in experimental analysis of metal deformation and in contacts. This neglect leads to degraded measurement accuracy of material properties. Therefore a need arises for material models that a priori incorporate inelasticity. These material models must be simple and comprehensive to have the highest impact in society. This thesis addresses three main sources of inelasticity, namely anelasticity and plasticity in metals and viscosity in contacts. Inelasticity is a dissipative mode of deformation that is mechanically recoverable for anelasticity and viscosity, and irrecoverable for plasticity. We connect the fundamental properties and structures of metallic and soft matter constituents with experimentally accessible measures. The presented models will aid in the development of materials with specific properties that meet the needs of industry.

Chapter 2 presents an analytical model of the tensile test tangent moduli and yield points for single-crystallite metals with spatially uniform and nonuniform dislocation distributions across slip systems. The moduli and the onset of plastic flow show a notable dependence on initial dislocation character, spatial dislocation distribution, and loading direction with respect to crystallographic orientations. An improved methodology accounts for elastic compressibility and anisotropy, and the geometric structure of crystal lattices when one measures dislocation network geometry in single metallic crystallites.

Chapter 3 contains a seamless, unified stress-strain treatment of dislocation-driven deformation. This treatment combines the three deformation mechanisms of elastic bond stretching, stable dislocation glide, and unstable dislocation glide. The model’s yield criterion connects the bowing out of local dislocation links and global dislocation multiplication. A semi-empirical relation is constructed for the evolution of the dislocation network structure with uniaxial loading.

Chapter 4 formulates a macromechanical model of the yield point phenomenon under invariant plane conditions. The heterogeneous stress state across the Lüders front and the plastic flow inside the Lüders band are accounted for. The Lüders band orientation with respect to the tensile direction is not unique; the orientation changes with material properties and tensile specimen geometry by the stress concentration at the front. The model serves to approximate constitutive parameters independent of the test conditions.

Chapter 5 elucidates the interplay between adhesion and roughness by modelling the retraction of rigid, wavy indenters from viscoelastic substrates. Viscoelasticity governs adhesive hysteresis across all loading rates, and even in the presence of roughness-induced mechanical instabilities. This confirms the central role that viscoelasticity must play in experimental measurements in the presence of adhesive interfaces in soft matter contacts.

Chapter 6 examines the static, quasi-static, and dynamic trajectories of a base-excited mass-spring-damper system in the presence of friction. The differences between the dynamic and the quasi-static solution in engineering problems with viscous, static, and dry friction are assessed. The omission of inertial contributions will under-predict dissipation at both low and high excitation frequencies. This chapter is a guide for future (multi-scale) numerical modelling efforts on adhesion and interface friction, and the hysteretic deformation of metals.

Chapter 7 is a general discussion on the impact of inelasticity in metals, that follows from Chapters 2 and 3, the measurement of the yield point phenomenon in Chapter 4, and numerical modelling of dissipative contacts in Chapters 5 and 6. The four models as presented in Chapters 2-6 are readily applicable in experimental measurements and future numerical models. The importance of accounting for inelasticity in experimental measurement and modelling of the yield strength in metals, and adhesive dissipation in soft matter contacts is emphasised. Finally, the state of the art in research on the three main sources of inelasticity and potential applications of the presented models are enumerated, which serve as starting points of future research.