A dislocation density based crystal plasticity — phase-field model is applied to investigate directional coarsening during creep in CMSX-4 Ni-based superalloys in the high temperature and low stress regime. Coherency between the ordered γ′ precipitates and the disordered γ channels prevents the generation of geometrically necessary dislocations, since the precipitate can be considered undeformable in the low stress regime. After coherency loss between the γ matrix phase and the γ′ precipitates the constraint against generation of geometrically necessary dislocations is relaxed, causing rotation of the crystal lattice under uniaxial load, known as “Schmid rotation”. As a consequence, the creep rate in the matrix increases, whereby degradation can be measured by the number density of geometrically necessary dislocations. The state of coherency loss is associated with the minimum creep rate in a creep experiment under constant load. The presented simulations start from a coherent γ′ precipitates distribution with random size and position generated during a precipitation heat treatment process. Simulations of N-type and P-type rafting under tensile and compressive load respectively are presented. The effect of coherency loss, coalescence of precipitates and lattice rotation due to generation of geometrically necessary dislocations is discussed in correlation with experimental findings. © 2023 Elsevier B.V.

The high-temperature austenite phase is the initial state of practically all technologically relevant hot forming and heat treatment operations in steel processing. The phenomena occurring in austenite, such as recrystallization or grain growth, can have a decisive influence on the subsequent properties of the material. After the hot forming or heat treatment process, however, the austenite transforms into other microstructural constituents and information on the prior austenite morphology are no longer directly accessible. There are established methods available for reconstructing former austenite grain boundaries via metallographic etching or electron backscatter diffraction (EBSD) which both exhibit shortcomings. While etching is often difficult to reproduce and strongly depend on the investigated steel’s alloying concept, EBSD acquisition and reconstruction is rather time-consuming. But in fact, though, light optical micrographs of steels contrasted with conventional Nital etchant also contain information about the former austenite grains. However, relevant features are not directly apparent or accessible with conventional segmentation approaches. This work presents a deep learning (DL) segmentation of prior austenite grains (PAG) from Nital etched light optical micrographs. The basis for successful segmentation is a correlative characterization from EBSD, light and scanning electron microscopy to specify the ground truth required for supervised learning. The DL model shows good and robust segmentation results. While the intersection over union of 70% does not fully reflect the model performance due to the inherent uncertainty in PAG estimation, a mean error of 6.1% in mean grain size derived from the segmentation clearly shows the high quality of the result. Copyright © 2022 Bachmann, Müller, Britz, Durmaz, Ackermann, Shchyglo, Staudt and Mücklich.

In this study the partially divorced eutectic microstructure of α-Mg and β-Mg17Al12 was investigated by electron backscatter diffraction, transmission electron microscopy, and phase-field modeling in hypoeutectic Mg-Al alloys. The orientation relationships between the individual eutectic α grains, eutectic β phase, and primary α grains were investigated. While the amount of eutectic morphology is primarily determined by the Al content, the in-depth microstructure analyses and the phase-field simulation suggest non-interactive nucleation and growth of eutectic α phase in the β phase grown on the interdendritic primary α dendrites. Also, phase-field simulations showed a preferred nucleation sequence where the β phase nucleates first and subsequently triggers the nucleation of eutectic α phase at the moving β phase solidification front, which supports the microstructural analysis results. © 2021

45° rafting of Ni-based superalloys has been investigated with the help of creep test simulations applying a strain gradient crystal plasticity model coupled to the multi-phase field method. This mode of rafting lies in between P- A nd N-type rafting modes. The model parameters are calibrated against experimental data for N-type rafting under high temperature and low stress creep condition. By increasing the stress level, the mixed-mode rafting of precipitates with a clear tendency toward formation of 45° rafts is observed. We show that the key factor for the occurrence of this type of rafting is the generation of highly localized creep strain of more than 10% due to non-homogeneous creep deformation in the form of slip bands. We have successfully captured the evolution of microstructure under high stress leading to production of localized shear bands. © 2020 Elsevier Ltd.

The role and effect of γ′ precipitate size on the mechanical properties of Ni-base single crystal superalloy is investigated. The underlying mechanisms are analyzed on the one hand with the help of experiments including hardness and creep tests, and on the other hand with the help of two different simulation approaches by taking the typical γ/γ′ microstructure into account. Simulations, based on the crystal plasticity finite element method (CPFEM) are carried out for the hardness tests, whereas simulations, based on the crystal plasticity coupled phase-field method (CPPFM) are carried out for the creep tests. The hardness test simulation results show that the hardness of material varies inversely with the size of γ′ precipitates for a given γ′ phase volume fraction and it varies directly with the volume fraction of γ′ precipitates for a given precipitate size. These results are qualitatively consistent with the experimental observations. The creep simulation results show that the refinement of γ′ precipitates with a certain volume fraction of precipitates leads to an improvement of creep resistance by delaying the plastic activity in the material. © 2020 Acta Materialia Inc.

The role of inclination dependence of grain boundary energy on the microstructure evolution and the orientation distribution of grain boundary planes during grain growth in polycrystalline materials is investigated by three-dimensional phase-field simulations. The anisotropic grain boundary energy model uses the description of the faceted surface structure of the individual crystals and constructs an anisotropic energy of solid-solid interface. The energy minimization occurs by the faceting of the grain boundary due to inclination dependence of the grain boundary energy. The simulation results for a single grain show the development of equilibrium shapes (faceted grain morphologies) with different families of facets which agrees well with the theoretical predictions. The results of grain growth simulations with isotropic and anisotropic grain boundary energy for cubic symmetry show that inclination dependence of grain boundary energy has a significant influence on the grain boundary migration, grain growth kinetics and the grain boundary plane distribution. It has been shown that the model essentially reproduces the experimental studies reported for NaCl and MgO polycrystalline systems where the anisotropic distribution of grain boundary planes has a peak for the low-index {100} type boundaries. © 2020 Acta Materialia Inc.

We present three-dimensional phase-field simulations of martensite microstructure formation in low-carbon steel. In this study, a full set of 24 Kurdjumov-Sachs symmetry variants of martensite is considered. Three different carbon compositions are investigated in order to reveal the effect of carbon content on the martensite microstructure formation. The simulations are performed using the finite strain framework which allows considering real martensite transformation strains. Using Neuber elasto-plastic approximation to the mechanical equilibrium solution, realistic stresses and strains can be obtained during martensite formation resulting in realistic mechanical driving forces for the transformation. The simulated microstructures are compared to experimental results for three carbon compositions. Good agreement between simulated and experimental results is achieved. © 2019 Acta Materialia Inc.

Rejuvenation of the initially hot isostatic pressing (HIP) heat-treated single-crystal Ni-base superalloy (SX) ERBO/1 was examined experimentally and via phase field simulation to establish rejuvenation treatments as a cost-effective alternative for another interval of service life. Creep was performed at 950 °C and 350 MPa, and the specimens were crept to 0.6 pct (creep rate minimum) or 2 pct strain, respectively. A slight coarsening of the γ/γ′ microstructure was observed experimentally and via simulation at 0.6 pct and rafting at 2 pct strain. The damaged microstructures were rejuvenated in a novel hot isostatic press that provides fast quenching rates before the same specimens were recrept under the same initial creep conditions. High-resolution microscopy proves that the rejuvenation re-establishes the original γ/γ′ microstructure in the dendrite core of the precrept specimens (0.6 and 2 pct). However, the interdendritic areas of the 2 pct precrept and rejuvenated specimen still contain elongated γ′ particles enwrapped by interfacial dislocation networks that survived the applied rejuvenation. The subsequent experimental and simulated creep tests after rejuvenation demonstrated that the creep behavior is only reproducible by the proposed rejuvenation for specimens that had crept until the end of the primary creep regime. © 2018 The Minerals, Metals & Materials Society and ASM International

The open-source software project OpenPhase allows the three-dimensional simulation of microstructural evolution using the multiphase field method. The core modules of OpenPhase and their implementation as well as their parallelization for a distributed-memory setting are presented. Especially communication and load-balancing strategies are discussed. Synchronization points are avoided by an increased halo-size, i.e. additional layers of ghost cells, which allow multiple stencil operations without data exchange. Load-balancing is considered via graph-partitioning and sub-domain decomposition. Results are presented for performance benchmarks as well as for a variety of applications, e.g. grain growth in polycrystalline materials, including a large number of phase fields as well as Mg–Al alloy solidification. Program summary Program Title: OpenPhase Program Files doi: http://dx.doi.org/10.17632/2mnv2fvkkk.1 Licensing provisions: GPLv3 Programming language: C++ Nature of problem: OpenPhase[1] allows the simulation of microstructure evolution during materials processing using the multiphase field method. In order to allow an arbitrary number of phase fields active parameter tracking is used, which can cause load imbalances in parallel computations. Solution method: OpenPhase solves the phase field equations using an explicit finite difference scheme. The parallel version of OpenPhase provides load-balancing using over-decomposition of the computational domain and graph-partitioning. Adaptive sub-domain sizes are used to minimize the computational overhead of the over-decomposition, while allowing appropriate load-balance. Additional comments including Restrictions and Unusual features: The distributed-memory parallelism in OpenPhase uses MPI. Shared-memory parallelism is implemented using OpenMP. The library uses C++11 features and therefore requires GCC version 4.7 or higher. [1] www.openphase.de © 2017 Elsevier B.V.

Ni-base superalloys are materials which are designed to resist extreme thermal and mechanical conditions. In this regard, an essential factor is their microstructure consisting of γ′ precipitates embedded in a γ matrix. The application of superalloys at high temperatures can however induce the topological phase inversion, where the γ′-phase topologically becomes the matrix phase, resulting in subpar material properties. In this work, the topological inversion is analyzed via experiment and phase-field simulation. The evolution of the microstructure has been quantified in the second generation single crystal Ni-base superalloy ERBO/1, which belongs to the family of CMSX-4, submitted to long-term aging at 1100° C for up to 250 h. Phase-field simulations carried out using a multi phase-field approach deliver insight into the microstructure evolution driven by the loss of coherency of the γ′ precipitates, which is induced by the accumulation of dislocations at the γ/γ′ interfaces. The obtained simulation results are in good agreement with the experimental results, and indicate that the mechanisms causing the topological inversion are linked to the accommodation of the lattice misfit, which enables coalescence and ripening of γ′ precipitates. © 2016 Acta Materialia Inc.

In this study we propose a unified multi-scale chemo-mechanical description of the BCT (Body-Centered Tetragonal) to BCC (Body-Centered Cubic) order-disorder transition in martensitic steel by adding the mechanical degrees of freedom to the standard CALPHAD (CALculation of PHAse Diagrams) type Gibbs energy description. The model takes into account external strain, the effect of carbon composition on the lattice parameter and elastic moduli. The carbon composition effect on the lattice parameters and elastic constants is described by a sublattice model with properties obtained from DFT (Density Functional Theory) calculations; the temperature dependence of the elasticity parameters is estimated from available experimental data. This formalism is crucial for studying the kinetics of martensite tempering in realistic microstructures. The obtained extended Gibbs energy description opens the way to phase-field simulations of tempering of martensitic steel comprising microstructure evolution, carbon diffusion and lattice symmetry change due to the ordering/disordering of carbon atoms under multiaxial load. © 2016 by the authors.

Martensitic steels form a material class with a versatile range of properties that can be selected by varying the processing chain. In order to study and design the desired processing with the minimal experimental effort, modeling tools are required. In this work, a full processing cycle from quenching over tempering to mechanical testing is simulated with a single modeling framework that combines the features of the phase-field method and a coupled chemo-mechanical approach. In order to perform the mechanical testing, the mechanical part is extended to the large deformations case and coupled to crystal plasticity and a linear damage model. The quenching process is governed by the austenite-martensite transformation. In the tempering step, carbon segregation to the grain boundaries and the resulting cementite formation occur. During mechanical testing, the obtained material sample undergoes a large deformation that leads to local failure. The initial formation of the damage zones is observed to happen next to the carbides, while the final damage morphology follows the martensite microstructure. This multi-scale approach can be applied to design optimal microstructures dependent on processing and materials composition. © 2016 by the authors.

A new simulation environment is developed to simulate the evolution of microstructure and the corresponding flow stress during rolling. An orientation dependent crystal plasticity hardening model is coupled to grain evolution-, recovery-and recrystallization kinetics within a phase field framework. Hardening and softening kinetics are treated consecutive to differentiate between individual effects. Simulation results are compared to hot compression tests at 1373 K with a strain rate of 1 s-1. © The Authors, published by EDP Sciences, 2016.

We present simulations of the nucleation and equiaxed dendritic growth of the primary hexagonal close-packed α-Mg phase followed by the nucleation of the β-phase in interdendritic regions. A zoomed-in region of a melt channel under eutectic conditions is investigated and compared with experiments. The presented simulations allow prediction of the final properties of an alloy based on process parameters. The obtained results give insight into the solidification processes governing the microstructure formation of Mg-Al alloys, allowing their targeted design for different applications. © 2015, The Minerals, Metals & Materials Society.

Phase-field simulations of the nucleation and growth of primary α-Mg phase as well as secondary, β-phase of a Mg-Al alloy are presented. The nucleation model for α- and β-Mg phases is based on the "free growth model" by Greer et al.. After the α-Mg phase solidification we study a divorced eutectic growth of α- and β-Mg phases in a zoomed in melt channel between α-phase dendrites. The simulated cooling curves and final microstructures of α-grains are compared with experiments. In order to further enhance the resolution of the interdendritic region a high-performance computing approach has been used allowing significant simulation speed gain when using supercomputing facilities. © Published under licence by IOP Publishing Ltd.

Abstract We investigate the ability of a multi-order parameter phase field model with obstacle potentials to describe grain boundary premelting in equilibrium situations. In agreement with an energetic picture we find that the transition between dry and wet grain boundaries at the bulk melting point is given by the threshold 2σsl=σgb, with σsl being the solid-melt interfacial energy and σgb the energy of a dry grain boundary. The predictions for premelting are confirmed by simulations using the phase field package OpenPhase. For the prediction of the kinetics of melting along grain boundaries in pure materials, taking into account the short ranged interactions which are responsible for the grain boundary premelting, a sharp interface theory is developed. It confirms that for overheated grain boundaries the melting velocity is reduced (increased) for non-wetting (wetting) grain boundaries. Numerical steady state predictions are in agreement with a fully analytical solution in a subset of the parameter space. Phase field simulations confirm the predictions of the sharp interface theory. © 2015 Elsevier B.V.

A mechanism is presented which opposes coalescence of γ′-precipitates in Ni-base superalloys. The mechanism is based on the non-linear behaviour of the elastic energy in γ-channels, caused by the misfit strain between matrix and precipitate, as a function of the channel width. Variation of the channel width causes a disjoining pressure dependent on the density of misfit dislocations. © 2015 Taylor & Francis.

Abstract A large-strain plasticity framework is set up for phase-field simulations at the mesoscopic scale. The approach is based on an Eulerian setting with remeshing after each time step to keep a fixed structured mesh. Rotations, as evaluated from the antisymmetric part of the deformation gradient tensor, are integrated to capture the process history. Special emphasis is also given to the homogenization of the diffuse interface region to ensure the Hadamard jump condition and 2-dimensional scaling of the interface. The approach is applied to deformation of a polycrystal. © 2015 Elsevier B.V.

Three-dimensional discrete dislocation dynamics (DDD) simulations in combination with the phase-field method are performed to investigate the influence of different realistic Ni-base single crystal superalloy microstructures with the same volume fraction of γ;precipitates on plastic deformation at room temperature. The phase-field method is used to generate realistic microstructures as the boundary conditions for DDD simulations in which a constant high uniaxial tensile load is applied along different crystallographic directions. In addition, the lattice mismatch between the γand γ;phases is taken into account as a source of internal stresses. Due to the high antiphase boundary energy and the rare formation of superdislocations, precipitate cutting is not observed in the present simulations. Therefore, the plastic deformation is mainly caused by dislocation motion in γ; matrix channels. From a comparison of the macroscopic mechanical response and the dislocation evolution for different microstructures in each loading direction, we found that, for a given γ;phase volume fraction, the optimal microstructure should possess narrow and homogeneous γ; matrix channels. © 2015 IOP Publishing Ltd Printed in the UK.

In this paper, we present the simulation of the eutectic phase transitions in the Pt–C system, in terms of both freezing and melting, using the multi-phase-field model. The experimentally obtained heat-extraction and -injection rates associated with the induction of freezing and melting are converted into the corresponding rates for microstructure-scale simulations. In spite of the extreme differences in the volume fractions of the FCC–Pt-rich phase on the one hand and graphite (C) on the other, satisfactory results for the kinetics of solidification and melting have been obtained, involving reasonable offsets in temperature, inducing freezing and melting, with respect to the equilibrium eutectic temperature. For freezing in the simulations, the needle/rod-like morphology, as experimentally observed, was reproduced for different heat extraction rates. The seemingly anomalous peak characterizing the simulated freezing curves is ascribed to the speed up of the solidification process due to the curvature effect. Similarly, a peak is observed in the experimental freezing curves, also showing up more clearly with increasing freezing rates. Melting was simulated starting from a frozen structure produced by a freezing simulation. The simulations reproduce the experimental melting curves and, together with the simulated freezing curves, help to understand the phase transition of the Pt–C eutectic. Finally, the effect of metallic impurities was studied. As shown for Au, impurities affect the morphology of the eutectic structure, their impact increasing with the impurity content, i.e., they can act as modifiers of the structure, as earlier reported for irregular eutectics. © 2015, Springer Science+Business Media New York.

This paper deals with the analysis of homogenization assumptions within phase field theories in a finite strain setting. Such homogenization assumptions define the average bulks energy within the diffusive interface region where more than one phase co-exist. From a physical point of view, a correct computation of these energies is essential, since they define the driving force of material interfaces between different phases. The three homogenization assumptions considered in this paper are: (a) Voigt/Taylor model, (b) Reuss/Sachs model, and (c) Khachaturyan model. It is shown that these assumptions indeed share some similarities and sometimes lead to the same results. However, they are not equivalent. Only two of them allow the computation of the individual energies of the co-existing phases even within the aforementioned diffusive interface region: the Voigt/Taylor and the Reuss/Sachs model. Such a localization of the averaged energy is important in order to determine and to subsequently interpret the driving force at the interface. Since the Voigt/Taylor and the Reuss/Sachs model are known to be relatively restrictive in terms of kinematics (Voigt/Taylor) and linear momentum (Reuss/Sachs), a novel homogenization approach is advocated. Within a variational setting based on (incremental) energy minimization, the results predicted by the novel approach are bounded by those corresponding to the Voigt/Taylor and the Reuss/Sachs model. The new approach fulfills equilibrium at material interfaces (continuity of the stress vector) and it is kinematically compatible. In sharp contrast to existing approaches, it naturally defines the mismatch energy at incoherent material interfaces. From a mathematical point of view, it can be interpreted as a partial rank-one convexification. © 2014 Elsevier Ltd.

In this study we present a large scale numerical simulation of γ-γ′ microstructure evolution in Ni-base superalloy using the multi-phase field method in three dimensions. We numerically simulated precipitation hardening heat treatment cycles. Large scale three dimensional simulations are necessary in order to get sufficient statistics for predicting the morphological evolution, average γ′ precipitate size, precipitates size distribution over time and ripening exponent for a given temperature and composition. A detailed analysis of obtained result is presented emphasising the effect of elastic interaction on the coarsening kinetics in Ni-base superalloy. The study is performed using the phase-field modelling library "OpenPhase" which is based on a multi-phase field multi-component model. © 2014 Owned by the authors, published by EDP Sciences.

Most solid-state phase transformations are accompanied by large deformations, stemming either from external load, transformation strains or plasticity. The consideration of such large deformations will affect the numerical treatment of such transformations. In this paper, we present a new scheme to embed large deformations in an explicit phase-field scheme and its implementation in the open-source framework OpenPhase. The suggested scheme combines the advantages of a spectral solver to calculate the mechanical boundary value problem in a small strain limit and an advection procedure to transport field variables over the calculation grid. Since the developed approach should be used for various sets of problems, e.g. simulations of thermodynamically driven phase transformations, the mechanic formulation is kept general. However, to ensure compatibility with phase-field methods using the concept of diffuse interface, the latter is treated with special care in the present work. © 2014 IOP Publishing Ltd.

We demonstrate that the distortion of a crystal, caused by secondary phase precipitates, can stabilize a solutal gradient around the precipitate. The gradient persists in the quasi-static state stabilized by the gradient of the elastic energy around the precipitate. The peak concentration at the interface between precipitate and matrix hereby is independent of the radius of the precipitate and no mechanism of ripening is active in an arrangement of precipitates of different size. The model offers an explanation of experimental observations of the anomalous stability of nano-precipitates in Al-Cu. © 2013 Taylor & Francis.

We present a simple Landau free energy functional for cubic-to-orthorhombic and cubic-to-monoclinic martensitic phase transformations. The functional is derived following group-subgroup relations between different martensitic phases - tetragonal, trigonal, orthorhombic and monoclinic - in order to fully capture the symmetry properties of the free energy of the austenite and martensite phases. The derived free energy functional is fitted to the elastic and thermodynamic properties of NiTi and NiTiCu shape memory alloys which exhibit cubic-to-monoclinic and cubic-to-orthorhombic martensitic phase transformations, respectively. © 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Phase-field modelling is maturing to become a universal tool for modelling microstructure evolution in materials science. In solidification applications it has been proven to give quantitative predictions. In solid state, however, the mechanisms of phase transformation and microstructure evolution in are much more involved due to the existence of mechanical interactions, high interface anisotropies, large densities of defects, and retarded kinetics of diffusion and growth. The paper gives an overview of actual developments in phase-field modelling of solid-state microstructure evolution and highlights necessary directions of future development in order to meet the challenge of quantitative predictions. © 2011 Elsevier Ltd. All rights reserved.