Duplex Stainless Steels (DSS) are used extensively in various industrial applications where the properties of both austenite and ferrite steels are required. Higher mechanical strength and superior corrosion resistance are the advantages of DSS. One of the main drawbacks for Duplex steels is precipitation of sigma phase and other intermetallic phases adversely affecting the mechanical strength and the corrosion behavior of the steels. The precipitation of these secondary phases and the associated brittleness can be due to improper heat treatment. The instability in the microstructure of Duplex stainless steels can be studied by understanding the phase transformations especially the ones involving sigma phase. To reduce the time and effort to be put in for experimental work, computational simulations are used to get an initial understanding on the phase transformations. The present thesis work is on the phase transformations involving sigma phase for Fe-Cr system and Fe-Cr-Ni system using theoretical approach in 1D and 2D geometries. A phase field model is implemented for the microstructural evolution in DSS in combination with thermodynamic data collected from the Thermo-Calc software. The Wheeler Boettinger McFadden (WBM) model is used for Gibbs energy interpolation of the system. FiPy- Finite volume PDE solver written in python is used to simulate the phase transformation conditions first in Fe-Cr system for ferrite-austenite and ferrite-sigma phase transformations. It is then repeated for Fe-Cr-Ni ternary system. In the present study a model was developed for deriving Gibbs energy expression for sigma phase based on the common tangent condition. This model can be used to describe composition constrained phases and stoichiometric phases using the WBM model in phase field modeling. Cogswell’s theory of using phase order variable instead of an interpolating polynomial in the expression for Gibbs energy of whole system is also tried.

In this thesis, a robust, reliable and efficient isogeometric analysis framework is presented that allows for an adaptive spatial discretization of non-linear and time-dependent multi-field problems. In detail, B\'ezier extraction of truncated hierarchical B-splines is proposed that allows for a strict element viewpoint, and in this way, for the application of standard finite element procedures. Furthermore, local mesh refinement and coarsening strategies are introduced to generate graded meshes that meet given minimum quality requirements. The different strategies are classified in two groups and compared in the adaptive isogeometric analysis of two- and three-dimensional, singular and non-singular problems of elasticity and the Poisson equation. Since a large class of boundary value problems is non-linear or time-dependent in nature and requires incremental solution schemes, projection and transfer operators are needed to transfer all state variables to the new locally refined or coarsened mesh. For field variables, two novel projection methods are proposed and compared to existing global and semi-local versions. For internal variables, two different transfer operators are discussed and compared in numerical examples. The developed analysis framework is than combined with the phase-field method. Numerous phase-field models are discussed including the simulation of structural evolution processes to verify the stability and efficiency of the whole adaptive framework and to compare the projection and transfer operators for the state variables. Furthermore, the phase-field method is used to develop an unified modelling approach for weak and strong discontinuities in solid mechanics as they arise in the numerical analysis of heterogeneous materials due to rapidly changing mechanical properties at material interfaces or due to propagation of cracks if a specific failure load is exceeded. To avoid the time consuming mesh generation, a diffuse representation of the material interface is proposed by introducing a static phase-field. The material in the resulting transition region is recomputed by a homogenization of the adjacent material parameters. The extension of this approach by a phase-field model for crack propagation that also accounts for interface failure allows for the computation of brittle fracture in heterogeneous materials using non-conforming meshes.

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Analyses of heat and mass transfer between different materials and phases are essential in numerous fundamental scientific problems and practical engineering applications, such as thermal and chemical transport in porous media, design of heat exchangers, dendritic growth during solidification, and thermal/mechanical analysis of additive manufacturing processes. In the numerical simulation, interface treatment can be further divided into sharp interface schemes and diffuse interface schemes according to the morphological features of the interface. This work focuses on the following subjects through computational studies: (1) critical evaluation of the various sharp interface schemes in the literature for conjugate heat and mass transfer modeling with the lattice Boltzmann method (LBM), (2) development of a novel sharp interface scheme in the LBM for conjugate heat and mass transfer between materials/phases with very high transport property ratios, and (3) development of a new diffuse-interface phase-field-lattice Boltzmann method (PFM/LBM) for dendritic growth and solidification modeling. For comparison of the previous sharp interface schemes in the LBM, the numerical accuracy and convergence orders are scrutinized with representative test cases involving both straight and curved geometries. The proposed novel sharp interface scheme in the LBM is validated with both published results in the literature as well as in-house experimental measurements for the effective thermal conductivity (ETC) of porous lattice structures. Furthermore, analytical correlations for the normalized ETC are proposed for various material pairs and over the entire range of porosity based on the detailed LBM simulations. In addition, we provide a modified correlation based on the SS420-air and SS316L-air metal pairs and the high porosity range for specific application. The present PFM/LBM model has several improved features compared to those in the literature and is capable of modeling dendritic growth with fully coupled melt flow and thermosolutal convection-diffusion. The applicability and accuracy of the PFM/LBM model is verified with numerical tests including isothermal, iso-solutal and thermosolutal convection-diffusion problems in both 2D and 3D. Furthermore, the effects of natural convection on the growth of multiple crystals are numerically investigated.

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The martensite and magnetic transformations in Ni-Fe-Ga ferromagnetic shape memory alloys are very sensitive to both alloy chemistry and thermal history. A series of Ni-Fe-Ga alloys near the prototype Heusler composition (X2YZ) were fabricated and homogenized at 1423 °K, and a Ni₅₃Fe₁₉Ga₂₈ alloy was subsequently annealed at various temperatures below and above the B2/L21 ordering temperature. Calorimetry and magnetometry were employed to measure the martensite transformation temperatures and Curie temperatures. Compositional variations of only a few atomic percent result in martensite start temperatures and Curie temperatures that differ by about 230 °K degrees and 35 °K degrees, respectively. Various one-hour anneals of the Ni₅₃Fe₁₉Ga₂₈ alloy shift the martensite start temperature and the Curie temperature by almost 70 °K degrees. Transmission electron microscopy investigations were conducted on the annealed Ni₅₃Fe₁₉Ga₂₈ alloy. The considerable variations in the martensite and magnetic transformations in these alloys are discussed in terms of microstructural differences resulting from alloy chemistry and heat treatments. The phase-field method has been successfully employed during the past ten years to simulate a wide variety of microstructural evolution in materials. Phase-field computational models describe the microstructure of a material by using a set of field variables whose evolution is governed by thermodynamic functionals and kinetic continuum equations. A two dimensional phase-field model that demonstrates the ferromagnetic shape memory effect in Ni2MnGa is presented. Free energy functionals are based on the phase-field microelasticity and micromagnetic theories; they account for energy contributions from martensite variant boundaries, elastic strain, applied stress, magnetocrystalline anisotropy, magnetic domain walls, magnetostatic potential, and applied magnetic fields. The time-dependent Ginzburg-Landau and Landau-Lifshitz kinetic continuum equations are employed to track the microstructural and magnetic responses in ferromagnetic shape memory alloys to applied stress and magnetic fields. The model results show expected microstructural responses to these applied fields and could be potentially utilized to generate quantitative predictions of the ferromagnetic shape memory effect in these alloys. / Ph. D.

The prediction of fracture is of utmost importance regarding the design of modern, specifically tailored engineering materials. These materials are often heterogeneous, \ie their properties vary in space. This can either be achieved purposefully by combining two or more constituents to profit from a more resilient composite material, or happen due to unavoidable imperfections. In any case, purely experimental assessment of failure is tedious and circuitous as soon as the structure of interest gets more complex, and the involvement of numerical models is inevitable. In this work, the phase-field approach to fracture is applied which is able to capture manifold crack phenomena inherently and sidesteps the need for remeshing by describing the crack as a continuous field. The phase-field model is extended to a fully diffuse incorporation of heterogeneities: A static order parameter smoothly transitions from one to the other bulk material constituent, while the weak, brittle interface is incorporated via a continuous fracture toughness distribution. The sharp interface jump conditions still hold for the diffuse representation since a partial rank-1 relaxation is employed in accordance with the unilateral contact condition of the phase-field model. Moreover, a compensation procedure ensures the independence of the interface fracture toughness from interface and phase-field length scales. The model is validated by a comparison to analytical results and the predictive power is demonstrated by the deduction of direction-dependent, effective fracture properties of heterogeneous microstructures. / Die Vorhersage des Bruchverhaltens ist für die Entwicklung moderner, speziell zugeschnittener technischer Werkstoffe von größter Bedeutung. Diese Materialien sind oft heterogen, d.h. ihre Eigenschaften variieren im Raum. Dies kann entweder absichtlich durch die Kombination zweier oder mehrerer Bestandteile erreicht werden, um von einem widerstandsfähigeren Verbundwerkstoff zu profitieren, oder durch unvermeidbare Imperfektionen geschehen. In jedem Fall ist eine rein experimentelle Versagensbewertung komplexer Strukturen mühsam und umständlich, und die Nutzung numerischer Modelle unvermeidlich. In dieser Arbeit werden Brüche mittels Phasenfeldmethode modelliert, wodurch vielfältige Rissphänomene erfasst werden können und die Notwendigkeit einer Neuvernetzung durch die Beschreibung des Risses als kontinuierliches Feld entfällt. Das Phasenfeldmodell wird um eine vollständig diffuse Einbindung von Heterogenitäten erweitert: Ein statischer Orderparameter beschreibt den glatten Übergang zwischen zwei Bestandteilen des Materials, während die geschwächte, spröde Grenzfläche durch eine kontinuierliche Bruchzähigkeitsverteilung eingebunden wird. Die scharfen Grenzflächensprungbedingungen gelten auch für die diffuse Darstellung, da eine partielle Rang-1 Relaxation in Übereinstimmung mit der unilateralen Rissflächenkontaktbedingung genutzt wird. Darüber hinaus gewährleistet ein Kompensationsverfahren die Unabhängigkeit der Grenzflächenbruchzähigkeit von inhärenten Längenskalen der Grenzfläche und des Rissphasenfelds. Das Modell wird durch einen Vergleich mit analytischen Ergebnissen validiert und die Vorhersagekraft wird durch die Ableitung richtungsabhängiger, effektiver Brucheigenschaften heterogener Mikrostrukturen demonstriert.