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A phase-field model is utilized to associate the solidification behavior of dendrites to the microscopic characteristics of the weld and to characterize the microstructure evolution in dissimilar pulsed laser welded Hastelloy X to 347 stainless steel alloy. The simulations reveal that the morphology of the dendrites during rapid solidification of the molten zone is affected by a melt flow and dilution level. The effect of melt flow as a result of Marangoni convection is modeled by Boussinesq approximation to adjust the concentration field around a developing dendrite, modifying its growth morphologies. The enhancement of hot cracking resistance is studied by adjusting the microstructure morphology through the possibility of the backfilling of the melt for the most efficient dendrite spacing, which was evaluated by correlation of the heat conduction problem and the phase-field model. Besides, the laser offset was estimated by finding the optimal chemical composition of the weld zone in the ternary Fe-Nieq-Creq system that affects the microstructure predicted by the phase-field model. The segregation of nanoparticle compounds analyzed by TEM in interdendritic regions is possible for primary dendrite spacing higher than 3 μm and consequently the solidification cracking susceptibility is increased.

A multi-scale model integrating a macro-scale finite element model and micro-scale phase field models is proposed to simulate the grain growth in the molten pool of laser beam welding of the Inconel 718 alloy. The calculated welding profile using the macro-scale model agrees well with experimental measurements. According to the calculated macro-scale temperature field, the temperature gradient G, the solidification rate R in the molten pool are calculated. The micro-scale phase field models of macro-scale G and R are coupled are used to simulate the grain growth in the molten pool. Simulated columnar grains are dendritic, which gives agreements with experimental observations. The simulation results and experimental measurements show that the columnar grains spacing increases slightly from the fusion edge to the weld. A power-law function of λ s ∝(G × R) −0.433 is found from the simulation results, which is similar to the power-law function of λ s ∝(G × R) −0.431 from the experimental measurements. Simulated microstructural patterns of the competitive growth match experimental findings well. The spacing of favorable oriented grains is a bit larger than that of unfavorable oriented grains in the competitive growth.

The Pore Formation and Mobility Investigation at the International Space Station provided information on the morphological evolution during remelting and directional solidification under microgravity conditions for Succinonitrile-0.24 wt% water binary alloys. Unlike the terrestrial experiments where the growth is affected by natural convection, constrained diffusive growth is observed in the microgravity experiments. This study aims to provide an experimental benchmark of dendritic growth applicable for validation of theoretical and numerical dendrite growth models. The results of the experiment were compared with the cellular automata and the phase field models, which are two classes of numerical methods widely used by scholars in the field of dendritic solidification, in both two and three dimensions. The resulting morphologies and tip velocities from the models were compared with the Pore Formation and Mobility Investigation experimental results. The combination of experimental and simulation results shows fair agreement and together can be used as a benchmark solution for tip velocity and evolution of dendritic microstructures.

Simple probabilistic models of fracture of 3D polycrystals involving transgranular and intergranular cracks are proposed, based on undamaged paths and appropriate percolation thresholds. In a second part, we propose theoretical extensions of phase field models for crack initiation and propagation to the case of locally heterogeneous and anisotropic fracture energy, and their implementation to get full field solutions by means of iterations of Fourier transforms to replace the standard finite element approach.

The phase field method has emerged as the most powerful approach for modeling microstructure evolutions resulting from phase transformations. This approach provides a consistent thermodynamic framework in which many phenomena can be strongly coupled, such as diffusion, elasticity, and plasticity. The method is therefore particularly suited for studying the microstructure formation and evolution in nickel-base superalloys. In this chapter, we present the strength and limitations of the method and illustrate how the phase field methods have contributed to the understanding of the microstructure evolution in these alloys.

This paper proposes an effective computational tool for brittle crack propagation problems based on a combination of a higher-order phase-field model and a non-conforming mesh using a NURBS-based isogeometric approach. This combination, as demonstrated in this paper, is of great benefit in reducing the computational cost of using a local refinement mesh and a higher-order phase-field, which needs higher derivatives of basis functions. Compared with other approaches using a local refinement mesh, the Virtual Uncommon-Knot-Inserted Master-Slave (VUKIMS) method presented here is not only simple to implement but can also reduce the variable numbers. VUKIMS is an outstanding choice in order to establish a local refinement mesh, i.e. a non-conforming mesh, in a multi-patch problem. A phase-field model is an efficient approach for various complicated crack patterns, including those with or without an initial crack path, curved cracks, crack coalescence, and crack propagation through holes. The paper demonstrates that cubic NURBS elements are ideal for balancing the computational cost and the accuracy because they can produce accurate solutions by utilising a lower degree of freedom number than an extremely fine mesh of first-order B-spline elements., Comment: 40 pages, 35 figures

In this paper, a quasi-brittle damage model for micro-architectural materials is presented within the framework of isogeometric analysis to exploit the high-order continuity of the non-uniform B-spline basis functions. The constitutive relation depends not only on the strain field, but also on their first and second strain gradient terms. The simplified second-gradient elasticity formulation from Mindlin’s theory is employed with corresponding micro-architecture-related length scales to capture the material nonlocality and size effects. The strain-based damage is modelled by a nonlocal independent field coupled to the displacement field. Influences of the two types of nonlocalities (manufactured micro-architectures and damage-induced micro-defects) on the response of structures, as well as the damage initiation and propagation, are analysed through numerical experiments. A formula to determine the micro-defect-related length scale from macroscopic measurements is proposed, boosting the accuracy and applicability of the model. In addition, relevant open problems and further developments of this damage model are discussed.

Due to renewed interest in uranium silicide compounds as a candidate for nuclear reactor fuels, there is a need for extensive investigations of their thermophysical properties as a function of temperature. In this work, we calculate the thermophysical properties of the U3Si2 compound within the framework of molecular dynamics (MD) using a semi-empirical modified Embedded-Atom Method (MEAM) potential and density functional theory (DFT). Thermal expansion, thermal conductivity, heat capacity, and elastic properties are presented as a function of temperature from 300 to 1800 K. The thermal conductivity of U3Si2 increases with temperature due to the electronic contribution while the phonon contribution decreases with increasing temperature. The phonon contribution to the thermal conductivity at 300 K is estimated at 2.03 W/mK and 1.41 W/mK using non-equilibrium molecular dynamics (NEMD) and equilibrium molecular dynamics (EMD), respectively. The electronic contribution is estimated to be 8.56 W/mK using the semi-classical Boltzmann transport theory at 300 K. Furthermore, we compared the thermal conductivity in two different crystallographic directions to shed light on the spatial anisotropy using NEMD and EMD methods. The inherent anisotropic thermophysical properties can be used to parametrize phase field models to incorporate anisotropic thermal conductivity and thermal expansion, allowing for a more accurate description of microstructural evolution under variable temperature and irradiation conditions.

This paper is aimed at the development of a coupled finite element-phase field modeling framework for simulating fracture processes in anisotropic elastic materials under finite deformation conditions with tension-compression asymmetry of crack phase. A novel Helmholtz free energy density is proposed in terms of a phase field order parameter dependent elastic Green-Lagrange strain tensor. It accommodates anisotropy of the material and unilateral conditions of material damage. The paper also addresses numerical instabilities, which adversely affect computational fracture simulations. Solutions based on viscous stabilization are suggested to effectively overcome the convergence issues of nonlinear FE solvers. Numerical examples are carried out to highlight different aspects of the developed framework. Numerical results with the proposed model are successfully compared with experimental results for validation. The model admits the unilateral conditions of the crack phase and predicts the correct crack kinking in mode II fracture. Special attention is drawn to numerical details affecting the efficiency and accuracy of the phase field models.

Microstructure evolution in the electrodes of solid oxide fuel cell is an important degradation mechanism which reduces active sites for redox reaction and the electric conductivity. Phase field models for microstructure evolution simulation are usually expensive for large scale simulations. In this work, a reduced-order coarsening model is developed using dynamic discrepancy reduced modeling, which reduces the model order by inserting Gaussian process stochastic functions into the dynamic equations of Ostwald ripening. The reduced order model has been calibrated on a dataset generated by a phase field model that has been well validated to experiments. A validating dataset has also been generated with which the model prediction show good agreement. This model is further applied to predict long term microstructure evolution in different SOFC electrodes. This work is the first attempt of building a degradation model of SOFC using data science techniques.

Solute trapping is an important phenomenon in rapid solidification of alloys, for which the continuous growth model (CGM) of Aziz et al. [1] is a popular sharp interface theory. By modulating the so-called anti-trapping current and using asymptotic analysis, we show how to quantitatively map the thin interface behavior of an ideal dilute binary alloy phase field model onto the CGM kinetics. We present the parametrizations that allow our phase field model to map onto the sharp interface kinetics of the CGM, both in terms of partition coefficient k ( V ) and kinetic undercooling. We also show that the mapping is convergent for different interface widths, both in transient and steady state simulations. Finally we present the effect that solute trapping can have on cellular growth in directional solidification. The presented treatment for solute trapping can be easily implemented in different phase field models, and is expected to be an important feature in future studies of quantitative phase field modeling in quasi-rapid solidification regimes, such as those relevant to metal additive manufacturing.

The results of molecular dynamics (MD) simulations of the crystallization process in one-component materials and solid solution alloys reveal a complex temperature dependence of the velocity of the crystal–liquid interface featuring an increase up to a maximum at 10–30% undercooling below the equilibrium melting temperature followed by a gradual decrease of the velocity at deeper levels of undercooling. At the qualitative level, such non-monotonous behaviour of the crystallization front velocity is consistent with the diffusion-controlled crystallization process described by the Wilson–Frenkel model, where the almost linear increase of the interface velocity in the vicinity of melting temperature is defined by the growth of the thermodynamic driving force for the phase transformation, while the decrease in atomic mobility with further increase of the undercooling drives the velocity through the maximum and into a gradual decrease at lower temperatures. At the quantitative level, however, the diffusional model fails to describe the results of MD simulations in the whole range of temperatures with a single set of parameters for some of the model materials. The limited ability of the existing theoretical models to adequately describe the MD results is illustrated in the present work for two materials, chromium and silicon. It is also demonstrated that the MD results can be well described by the solution following from the hodograph equation, previously found from the kinetic phase-field model (kinetic PFM) in the sharp interface limit. The ability of the hodograph equation to describe the predictions of MD simulation in the whole range of temperatures is related to the introduction of slow (phase field) and fast (gradient flow) variables into the original kinetic PFM from which the hodograph equation is obtained. The slow phase-field variable is responsible for the description of data at small undercoolings and the fast gradient flow variable accounts for local non-equilibrium effects at high undercoolings. The introduction of these two types of variables makes the solution of the hodograph equation sufficiently flexible for a reliable description of all nonlinearities of the kinetic curves predicted in MD simulations of Cr and Si. This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

International audience; Phase-field models of brittle fracture can be regarded as gradient damage models including an intrinsic internal length. This length determines the stability threshold of solutions with homogeneous damage and thus the strength of the material, and is often tuned to retrieve the experimental strength in uniaxial tensile tests. In this paper, we focus on multiaxial stress states and show that the available energy decompositions, introduced to avoid crack interpenetration and to allow for unsymmetric fracture behavior in tension and compression, lead to multiaxial strength surfaces of different but fixed shapes. Thus, once the length scale is tailored to recover the experimental tensile strength, it is not possible to match the experimental compressive or shear strength. We propose a new energy decomposition that enables the straightforward calibration of a multi-axial failure surface of the Drucker-Prager type. The new decomposition, which hinges upon the theory of structured deformations, encompasses the volumetric-deviatoric and the no-tension models as special cases. Preserving the variational structure of the model, it includes an additional free parameter that can be calibrated based on the experimental ratio of the compressive to the tensile strength (or, if possible, of the shear to the tensile strength), as successfully demonstrated on two data sets taken from the literature.

Materials 14(7), 1787 (2021). doi:10.3390/ma14071787 special issue: "Special Issue "Strong Coupling of Thermo-Chemical and Thermo-Mechanical States in Applied Materials" / Special Issue Editor: Prof. Ingo Steinbach, Guest Editor", Published by MDPI, Basel

We propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a ‘generalized Swift–Hohenberg equation’—a second-grade phase-field equation—and its conserved version, the ‘generalized phase-field crystal equation’—a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions.

Materials / Materials Diversity Preservation International 14(3), 654 (2021). doi:10.3390/ma14030654 special issue: "Special Issue "Strong Coupling of Thermo-Chemical and Thermo-Mechanical States in Applied Materials" / Special Issue Editor: Prof. Ingo Steinbach, Guest Editor", Published by MDPI, Basel

Abstract In order to approximate Gibbs energy functions, a semi-automated framework is introduced for binary and ternary material systems, using Calphad databases. To generate Gibbs energy formulations by means of second-order polynomials, the framework includes a precise approach. Furthermore, an optional extensional step enables the modeling of systems in which a direct generation leads to the unsatisfactory results in the representation of the thermodynamics. Furthermore, an optional extensional step enables the modeling of systems, in which a direct generation leads to the unsatisfactory results, when representing the thermodynamics. Within this extension, the commonly generated functions are modified to satisfy the equilibrium conditions in the observed material systems, leading to a better correlation with thermodynamic databases. The generated Gibbs energy formulations are verified by recalculating the equilibrium concentrations of the phases and rebuilding the phase diagrams in the considered concentration and temperature ranges, prior to the simulation studies. For all comparisons, a close match is achieved between the results and the Calphad databases. As practical examples of the method, phase-field simulation studies for the directional solidification of the binary – and the ternary – eutectic systems are performed. Good agreements between the simulation results and the reported theoretical and experimental studies from literature are found, which indicates the applicability of the presented approaches. Graphical Abstract

The phase field paradigm, in combination with a suitable variational structure, has opened a path for using Griffith's energy balance to predict the fracture of solids. These so-called phase field fracture methods have gained significant popularity over the past decade, and are now part of commercial finite element packages and engineering fitness- for-service assessments. Crack paths can be predicted, in arbitrary geometries and dimensions, based on a global energy minimization-without the need for ad hoc criteria. In this work, we review the fundamentals of phase field fracture methods and examine their capabilities in delivering predictions in agreement with the classical fracture mechanics theory pioneered by Griffith. The two most widely used phase field fracture models are implemented in the context of the finite element method, and several paradigmatic boundary value problems are addressed to gain insight into their predictive abilities across all cracking stages; both the initiation of growth and stable crack propagation are investigated. In addition, we examine the effectiveness of phase field models with an internal material length scale in capturing size effects and the transition flaw size concept. Our results show that phase field fracture methods satisfactorily approximate classical fracture mechanics predictions and can also reconcile stress and toughness criteria for fracture. The accuracy of the approximation is however dependent on modelling and constitutive choices; we provide a rationale for these differences and identify suitable approaches for delivering phase field fracture predictions that are in good agreement with well-established fracture mechanics paradigms. This article is part of a discussion meeting issue 'A cracking approach to inventing new tough materials: fracture stranger than friction'., Royal Commission for the 1851 Exhibition

Phase field models for fracture are energy-based and employ a continuous field variable, the phase field, to indicate cracks. The width of the transition zone of this field variable between damaged and intact regions is controlled by a regularization parameter. Narrow transition zones are required for a good approximation of the fracture energy which involves steep gradients of the phase field. This demands a high mesh density in finite element simulations if 4-node elements with standard bilinear shape functions are used. In order to improve the quality of the results with coarser meshes, exponential shape functions derived from the analytic solution of the 1D model are introduced for the discretization of the phase field variable. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the fracture field. Unfortunately, lower-order Gauss-Legendre quadrature schemes, which are sufficiently accurate for the integration of bilinear shape functions, are not sufficient for an accurate integration of the exponential shape functions. Therefore in this work, the numerical accuracy of higher-order Gauss-Legendre formulas and a double exponential formula for numerical integration is analyzed., Projekt DEAL

The phase-field method is increasingly used for studying grain growth in metallic alloys. Such an approach offers a thermodynamically consistent framework for studying microstructure evolution during grain growth without the need to explicitly track the interface positions. However, the numerical solution of the models requires a grid spacing much smaller than the interface width. This leads to computationally very intensive simulations, especially when a large number of grains is required to accurately measure statistical quantities. The recently proposed S-PFM approach provides a new inherently discrete formulation of phase field models where the interface width can be as small as the grid spacing, thus drastically improving the numerical performances of the method. Here, we show that this approach can be extended to a multi-phase field model for ideal grain growth. Then, we perform two dimensional simulations to analyse in detail the kinetics of both grain boundaries and triple junctions. We compare our model to a classical phase field formulation and we demonstrate that, for a prescribed accuracy, the memory requirement and simulation times are both reduced by a factor of 4 D , where D is the space dimension. Finally, we perform a large-scale simulation of grain growth in two dimensions and show that the method is able to reveal the specific shape of the grain size distribution in the scale-invariant regime displaying two peaks around the mean grain size and quantitative measurements of the underlying topological classes are performed.

Ferroelectric phase field models based on the Ginzburg–Landau–Devonshire theory are characterized by a large number of material parameters with problematic physical interpretation. In this study, we systematically address the relationship between these parameters and the main properties of ferroelectric domain walls. A variational approach is used to derive closed form solutions for the polarization fields at the phase transition regions as well as for the propagation velocities of the domain walls. Introducing a modified set of material parameters, which appropriately scales different contributions to the free energy, we are able to accurately calibrate these parameters based on domain wall thickness and energy of both 180$$^\circ $$∘and 90$$^\circ $$∘domain walls. Moreover, the mobility parameter appearing in the Ginzburg–Landau evolution equation can be accurately calibrated based on the propagation velocity of the domain walls.

International audience; In this paper, we give a formulation of distributed parameter systems with a moving diffuse interface using the Port Hamiltonian formalism. For this purpose, we suggest to use the phase field modeling approach. In the first part we recall the phase field models, in particular the Cahn-Hilliard and Allen-Cahn equations, and show that they may be expressed in terms of a dissipative Hamiltonian system. In the second part we show how this Hamiltonian model may be extended to a Boundary Port Hamiltonian System and illustrate the construction on the example of crystallization.

Predicting crack growth in structural components is a computationally intensive process. Phase field models of crack growth reduce the computational complications associated with singularities, and allow finite element predictions of crack propagation without remeshing. A novel approach to derive governing equations based on a Lagrangian density is proposed and the phase evolution is shown to be governed by a diffusion type equation with a source term. We correlate the phase to the micromechanical response. The model was incorporated in a finite element code and used to predict crack growth phenomena including (1) values of critical stress, (2) crack path, and (3) crack bifurcation. It was shown that the values of the critical stresses can be accurately estimated by the change in the phase field crack tip velocity or by the use of energy contour integral. The phase field crack analysis is compared against LEFM results.

We propose a new phase field model of dynamic brittle fracture, in which material damage evolves according to a hyperbolic partial differential equation. This model can be stably discretized using explicit time integration, without imposing crippling time step restrictions with refinement in space. The model is derived from microforce balance by including effects of microscopic inertia. Quantitative predictions of the proposed model differ from those of parabolic and elliptic phase field models in that there is a mild rate-toughening effect, but major qualitative solution features are essentially the same. We compute finite element approximations of solutions to several dynamic fracture scenarios to support these claims. Part II of this series will incorporate the proposed model into a hybrid isogeometric–meshfree framework for air-blast–structure interaction and provide further demonstrations of the model’s physical and numerical properties.

Unless corrected by so called anti-trapping currents, phase field models of solidification display a dependence upon the diffuse interface width, δ, used in the simulation. This is most commonly manifest as a reduction in solute partitioning, which is both growth velocity and interface width dependent, resulting in a serious impediment to quantitative simulation. However, such anti-trapping currents are often restricted to very simple materials thermodynamics, appropriate only to dilute ideal solutions. Here we propose a form of the anti-trapping current which can be implemented for arbitrary thermodynamics, including both Redlich-Kister solution phases and sub-lattice models for intermetallic growth. The effect of the new anti-trapping current is illustrated with respect to Pb dendrites growing from a Pb-Sn melt containing either 25% or 30% Sn. The new anti-trapping current is shown to render the solutions independent of the diffuse interface width both with regard to solute partitioning and other growth metrics such as solidification velocity and dendrite tip radius.

A thermodynamically consistent phase field model for crack propagation is analyzed. The thermodynamic driving force for the crack propagation is derived based on the laws of thermodynamics. The Helmholtz free energy satisfies the thermodynamic equilibrium and instability conditions for the crack propagation. Analytical solutions for the Ginzburg–Landau equation including the surface profile and the estimation of the kinetic coefficient are found. It is shown how kinetic coefficient affects the local stress field. The local critical stress for the crack propagation is calibrated with the theoretical strength which gives the value of the crack surface width. The finite element method is utilized to solve the complete system of crack phase field and mechanics equations. The analytical expressions for the crack surface profile and the crack tip velocity are verified with the numerical solutions. It is shown that under mode I cracking and proper calibration of parameters, phase field models always agree with Griffith’s theory.

We present the second set of benchmark problems for phase field models that are being jointly developed by the Center for Hierarchical Materials Design (CHiMaD) and the National Institute of Standards and Technology (NIST) along with input from other members in the phase field community. As the integrated computational materials engineering (ICME) approach to materials design has gained traction, there is an increasing need for quantitative phase field results. New algorithms and numerical implementations increase computational capabilities, necessitating standard problems to evaluate their impact on simulated microstructure evolution as well as their computational performance. We propose one benchmark problem for solidification and dendritic growth in a single-component system, and one problem for linear elasticity via the shape evolution of an elastically constrained precipitate. We demonstrate the utility and sensitivity of the benchmark problems by comparing the results of (1) dendritic growth simulations performed with different time integrators and (2) elastically constrained precipitate simulations with different precipitate sizes, initial conditions, and elastic moduli. These numerical benchmark problems will provide a consistent basis for evaluating different algorithms, both existing and those to be developed in the future, for accuracy and computational efficiency when applied to simulate physics often incorporated in phase field models.

Even as a commercial cathode material, LiFePO4 remains of tremendous research interest for understanding Li intercalation dynamics. The partially lithiated material spontaneously separates into Li-poor and Li-rich phases at equilibrium. Phase segregation is a surprising property of LiFePO4 given its high measured rate capability. Previous theoretical studies, aiming to describe Li intercalation in LiFePO4, include both atomic-scale density functional theory (DFT) calculations of static Li distributions and entire-particle-scale phase field models, based upon empirical parameters, studying the dynamics of the phase separation. Little effort has been made to bridge the gap between these two scales. In this work, DFT calculations are used to fit a cluster expansion for the basis of kinetic Monte Carlo calculations, which enables long time scale simulations with accurate atomic interactions. This atomistic model shows how the phases evolve in LixFePO4 without parameters from experiments. Our simulations revea...

Solidification processes are present in a wide range of manufacturing methods and applications, from metallurgy to food processing. In recent years, Phase Field models have been increasingly used to simulate and predict the formation and evolution of material microstructure and phase change interfacial kinetics. However, these methods usually lead to computationally involved numerical schemes, revealing the need for more efficient computational solutions. In this work, two different model reduction techniques, the Laplacian Spectral Decomposition and the Proper Orthogonal Decomposition, have been employed for model reduction of the Kobayashi model, a non-linear solidification Phase-field model. The performance of both low-order models has been illustrated and compared using a range of undercooling and seeding conditions. Results obtained accurately represent the behaviour of the full system, showing the potential of reduced order approaches for the modelling of complex interfacial systems.

A formal asymptotic analysis of two classes of phase field models for void growth and coarsening in irradiated solids has been performed to assess their sharp-interface kinetics. It was found that the sharp interface limit of type B models, which include only point defect concentrations as order parameters governed by Cahn-Hilliard equations, captures diffusion-controlled kinetics. It was also found that a type B model reduces to a generalized one-sided classical Stefan problem in the case of a high driving thermodynamic force associated with the void growth stage, while it reduces to a generalized one-sided Mullins-Sekerka problem when the driving force is low in the case of void coarsening. The latter case corresponds to the famous rate theory description of void growth. Type C models, which include point defect concentrations and a non-conserved order parameter to distinguish between the void and solid phases and employ coupled Cahn-Hilliard and Allen-Cahn equations, are shown to represent mixed diffusion and interfacial kinetics. In particular, the Allen-Cahn equation of model C reduces to an interfacial constitutive law representing the attachment and emission kinetics of point defects at the void surface. In the limit of a high driving force associated with the void growth stage, a type C model reduces to a generalized one-sided Stefan problem with kinetic drag. In the limit of low driving forces characterizing the void coarsening stage, however, the model reduces to a generalized one-sided Mullins-Sekerka problem with kinetic drag. The analysis presented here paves the way for constructing quantitative phase field models for the irradiation-driven nucleation and growth of voids in crystalline solids by matching these models to a recently developed sharp interface theory.

We have developed a data assimilation (DA) methodology based on the ensemble Kalman filter (EnKF) for estimating unknown parameters involved in a phase-field model from observational/experimental data. The DA methodology based on Bayesian statistics is able to estimate parameters by incorporating observational/experimental data into the phase-field model and evaluate the uncertainty of the estimated parameters. In this paper, we apply the EnKF-based DA method to estimate the phase-field mobility for a phase-field simulation of the isothermal austenite-to-ferrite transformation in a Fe–C–Mn alloy. Our DA method is validated through numerical experiments called “twin experiments” to verify that the DA method can estimate a priori assumed-true phase-field mobility from synthetic observational data. The results of the twin experiments using various initial phase-field mobilities show that our DA methodology can successfully estimate the true phase-field mobility, even when the initial value largely deviates from the true value. Furthermore, our DA method reveals the sampling interval for observational data necessary to accurately estimate the parameter and its uncertainty.

We present a new approach for incorporating CALPHAD thermodynamics in phase field models using linear spline functions. In this method, we discretize an n-component composition space with a mesh of n-1 dimensional elements that have linear shape/interpolation functions, therefore representing any thermodynamic quantity as a linear function of composition. Following this approach, the solution of auxiliary solute compositions in phase field simulations is straightforward and significantly faster compared to standard Redlich-Kister polynomials based method. At the same time, we obtain high accuracy in solutions when compared with reference calculations where excess energy is represented using Redlich-Kister polynomials. The salient features of the model are highlighted through examples of α / β phase transformation in a ternary Ti-Al-V diffusion couple.

In the present study, the multiphase volume distribution problem, where there can be an arbitrary number of phases, is addressed using a consistent and conservative volume distribution algorithm. The proposed algorithm satisfies the summation constraint, the conservation constraint, and the \textit{consistency of reduction}. The first application of the volume distribution algorithm is to determine the Lagrange multipliers in multiphase Phase-Field models that enforce the mass conservation, and a multiphase conservative Allen-Cahn model that satisfies the \textit{consistency of reduction} is developed. A corresponding consistent and conservative numerical scheme is developed for the model. The multiphase conservative Allen-Cahn model has a better ability than the multiphase Cahn-Hilliard model to preserve under-resolved structures. The second application is to develop a numerical procedure, called the boundedness mapping, to map the order parameters, obtained numerically from a multiphase model, into their physical interval, and at the same time to preserve the physical properties of the order parameters. Along with the consistent and conservative schemes for the multiphase Phase-Field models, the numerical solutions of the order parameters are reduction consistent, conservative, and bounded, which are theoretically analyzed and numerically validated. Then, the multiphase Phase-Field models are coupled with the momentum equation by satisfying the \textit{consistency of mass conservation} and the \textit{consistency of mass and momentum transport}, thanks to the consistent formulation. It is demonstrated that the proposed model and scheme converge to the sharp-interface solution and are capable of capturing the complicated multiphase dynamics even when there is a large density and/or viscosity ratio., The is an accepted manuscript

This article describes an approach to combine single phase thermodynamic potentials into a multiphase composite potential suitable for integration in a multicomponent phase-field model. The composite potential avoids implicit interfacial energy contributions by starting from a grand potential formulation. The method is made explicit by expanding the minimiser resulting from the Legendre transform between grand and Helmholtz potentials about a known equilibrium state. The resulting composite function is explicit, reproduces the equilibrium states exactly, and is smooth such that it can be differentiated to provide the driving forces for mass transport and phase change in a thermodynamically self-consistent manner. The model is demonstrated by simulating a sequence of phase transformations for intermetallic growth in Al-Mg interdiffusion for advanced nuclear research reactor fuel.

A phase field model of fracture that accounts for anisotropic material behavior at small and large deformations is outlined within this work. Most existing fracture phase field models assume crack evolution within isotropic solids, which is not a meaningful assumption for many natural as well as engineered materials that exhibit orientation-dependent behavior. The incorporation of anisotropy into fracture phase field models is for example necessary to properly describe the typical sawtooth crack patterns in strongly anisotropic materials. In the present contribution, anisotropy is incorporated in fracture phase field models in several ways: (i) Within a pure geometrical approach, the crack surface density function is adopted by a rigorous application of the theory of tensor invariants leading to the definition of structural tensors of second and fourth order. In this work we employ structural tensors to describe transverse isotropy, orthotropy and cubic anisotropy. Latter makes the incorporation of second gradients of the crack phase field necessary, which is treated within the finite element context by a nonconforming Morley triangle. Practically, such a geometric approach manifests itself in the definition of anisotropic effective fracture length scales. (ii) By use of structural tensors, energetic and stress-like failure criteria are modified to account for inherent anisotropies. These failure criteria influence the crack driving force, which enters the crack phase field evolution equation and allows to set up a modular structure. We demonstrate the performance of the proposed anisotropic fracture phase field model by means of representative numerical examples at small and large deformations.

The stagnant and inverse transformation stages during cyclic phase transformation in the two-phase region “ferrite + austenite” (intercritical region) of Fe-C-Mn and Fe-C-Mn-Si alloys were investigated using the non-equilibrium multi-phase-field (NEMPF) model, i.e. an MPF model with finite interface dissipation. We showed that if we used large interfacial permeability parameters that characterize the partitioning rate of solute atoms in the ferrite/austenite interface, the NEMPF model predicted the same transformation behavior as that simulated using the MPF model based on the assumption of the parallel tangent law. The simulation demonstrated that the inverse transformation stage, in which ferrite-to-austenite transformation proceeds despite decreasing temperature, could be naturally described by the NEMPF model coupled with the CALPHAD-based thermodynamic database. This study also elucidated a switching of the polarities of Mn and Si spike formed at the ferrite/austenite interface governed the driving force of the austenite-to-ferrite transformation and caused the stagnant stage where the phase transformation was suppressed. Furthermore, the impact of the amount of Mn and Si atoms on the stagnant stage length was examined using the NEMPF model. The simulations revealed that increasing the concentration of Mn and Si extended the stagnant stage, with a more pronounced effect with Mn rather than Si atoms.

Surface roughening is essential for the design of new nanostructures. In this paper, a continuum model based on phase field theory is developed to study the surface roughening in heteroepitaxial structures. A long-range order parameter is introduced to represent a diffusional free surface profile. By minimizing the system total free energy under constraints, the coupled field equations of quasi-static mechanical equilibrium and diffusion-controlled surface evolution are obtained. In contrast to the existing phase field models of free surface which are mostly based on the analytical Green’s function, the Galerkin finite element formulation is derived in this paper to solve the weak form of the coupled field equations. The advantage is that it is very convenient to deal with the elastic inhomogeneities and complex boundary conditions without additional complications. The proposed model is applied to study the nonlinear surface morphology evolution of heteroepitaxial thin films. The simulation shows that the heteroepitaxial film with an initial random fluctuation develops to form islands and coarsens during annealing. The kinetics of coarsening and the calculated average wavelength are consistent with the existing experimental results. It also shows that the initial island array formed during annealing consists of islands with similar size, shape and spacing. The influences of film thickness and misfit strain on the formation of initial island array are studied.

Phase-field models for crack propagation enable the simulation of complex crack patterns without complex and expensive tracking and remeshing as cracks grow. In the setting without inertia, the crack evolution is obtained from a variational energetic starting point, and leads to an equation for the order parameter coupled to elastostatics. Careful mathematical analysis has shown that this is consistent with the Griffith model for fracture. Recent efforts to include inertia in this formulation have replaced elastostatics by elastodynamics. In this brief note, we examine the elastodynamic augmentation, and find that it effectively causes the Griffith surface energy to depend on the velocity of the crack. That is, considering two identical specimens that are each fractured by a single crack that grows at different velocities in the two specimens, it is expected that the final equilibrium configurations are nominally identical; however, the phase-field fracture models augmented with elastodynamics achieve final configurations—in particular, the Griffiths surface energy contributions—that depend on the crack velocity. The physical reason is that the finite relaxation time for the stresses in the elastodynamic setting enables the cracked region to widen, beyond the value observed in the quasistatic setting. Once the crack widens, the “no-healing” condition prevents it from relaxing even after the specimen reaches equilibrium. In phase-field models, crack width in the reference configuration is unrelated to the physical opening of the crack but is instead a measure of Griffiths surface energy. This observation suggests that elastodynamic phase-field fracture models should not be used in settings where the crack velocity is large.

During solid–solid precipitation, interface free energy anisotropy is known to drive faceting of precipitates. In this paper, using a recently developed phase field formulation based on higher order tensor terms, we develop and implement a family of phase field models and indicate the parameter choices which lead to faceted precipitate morphologies. We also indicate how to choose the parameters given either the known precipitate morphology or the interfacial free energy anisotropy. Specifically, we study the faceting of precipitates in systems with cubic and hexagonal anisotropies; in 2 and 3D implementation of our phase field model, the precipitates do show facets in accordance with the Wulff plot – including cases where the Wulff plot predicts facets made up of more than one family of planes. We also indicate the possible extensions of our model to study other problems of interest.

Quantitative phase-field models have been developed as feasible computational tools for solving the free-boundary problem in solidification processes. These models are constructed with some polynomials of the phase-field variable that describe variations of the physical quantities inside the diffuse interface. The accuracy of the simulation depends on the choice of the polynomials and such dependence is indispensable for high-performance computing and valuable for extending the range of applications of the model to several physical systems. However, little is known about the dependence of the accuracy on the choice of the polynomials. In this study, numerical testing is carried out for quantitative phase-field models with extensive sets of polynomials (24 different models) for isothermal solidification in binary alloys. It is demonstrated in two-dimensional simulations of dendritic growth that a specific set of polynomials must be employed to achieve high accuracy in the models with double-well and double-obstacle potentials. Both types of model with the best set of polynomials yield almost the same numerical accuracy. (C) 2017 Elsevier Inc. All rights reserved.

Complex microstructure changes occur in nuclear fuel and structural materials due to the extreme environments of intense irradiation and high temperature. This paper evaluates the role of the phase field method in predicting the microstructure evolution of irradiated nuclear materials and the impact on their mechanical, thermal, and magnetic properties. The paper starts with an overview of the important physical mechanisms of defect evolution and the significant gaps in simulating microstructure evolution in irradiated nuclear materials. Then, the phase field method is introduced as a powerful and predictive tool and its applications to microstructure and property evolution in irradiated nuclear materials are reviewed. The review shows that (1) Phase field models can correctly describe important phenomena such as spatial-dependent generation, migration, and recombination of defects, radiation-induced dissolution, the Soret effect, strong interfacial energy anisotropy, and elastic interaction; (2) The phase field method can qualitatively and quantitatively simulate two-dimensional and three-dimensional microstructure evolution, including radiation-induced segregation, second phase nucleation, void migration, void and gas bubble superlattice formation, interstitial loop evolution, hydrate formation, and grain growth, and (3) The Phase field method correctly predicts the relationships between microstructures and properties. The final section is dedicated to a discussion of the strengths and limitations of the phase field method, as applied to irradiation effects in nuclear materials.

We analyze a control problem for a phase field system modeling the solidification process of materials that allow two different types of crystallization coupled to a Navier–Stokes system and a nonlinear heat equation, with a reduced number of controls. We prove that this system is locally exactly controllable to suitable trajectories, with controls acting only on the motion and heat equations.

Recent theoretical and computational progress has led to unprecedented understanding of symmetry-breaking instabilities in 2D dynamic fracture. At the heart of this progress resides the identification of two intrinsic, near crack tip length scales -- a nonlinear elastic length scale $\ell$ and a dissipation length scale $\xi$ -- that do not exist in the classical theory of cracks. In particular, it has been shown that at a high propagation velocity $v$, cracks in 2D brittle materials undergo an oscillatory instability whose wavelength varies linearly with $\ell$, and at yet higher propagation velocities and larger loading levels, a tip-splitting instability emerges, both in agreements with experiments. In this paper, using phase-field models of brittle fracture, we demonstrate the following properties of the oscillatory instability: (i) It exists also in the absence of near-tip elastic nonlinearity, i.e. in the limit $\ell\!\to\!0$, with a wavelength determined by the dissipation length scale $\xi$. This result shows that the instability crucially depends on the existence of an intrinsic length scale associated with the breakdown of linear elasticity near crack tips, independently of whether the latter is related to nonlinear elasticity or to dissipation. (ii) It is a supercritical Hopf bifurcation, featuring a vanishing oscillations amplitude at onset. (iii) It is largely independent of the fracture energy $\Gamma(v)$ that is controlled by a dissipation time scale. These results substantiate the universal nature of the oscillatory instability of ultra-high speed cracks in 2D. In addition, we provide evidence indicating that the ultra-high velocity tip-splitting instability is controlled by the limiting rate of elastic energy transport inside the crack tip region. Finally, we describe in detail the numerical implementation scheme of the employed phase-field fracture approach., Comment: Main changes: a new Appendix C (and Fig. C.1) and new data on tip-splitting angles

“This is a post-peer-review, pre-copyedit version of an article published in Computational mechanics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00466-020-01841-1”. A new adaptive refinement strategy for phase-field models of brittle fracture is proposed. The approach provides a computationally efficient solution to the high demand in spatial resolution of phase-field models. The strategy is based on considering two types of elements: h-refined elements along cracks, where more accuracy is needed to capture the solution, and standard elements in the rest of the domain. Continuity between adjacent elements of different type is imposed in weak form by means of Nitsche's method. The weakly imposition of continuity leads to a very local refinement in a simple way, for any degree of approximation and both in 2D and 3D. The performance of the strategy is assessed for several scenarios in the quasi-static regime, including coalescence and branching of cracks in 2D and a twisting crack in 3D.

Phase-field models for fracture have gained exceptional popularity in the last couple of decades and have been extended into areas well beyond brittle quasi-static fracture propagation to ductile, dynamic, or hydraulic fracturing. Despite the significant theoretical advancements in these more complex physical settings, little attention has been paid to the quantification of crack opening displacement to date as most applications do not explicitly require the crack opening displacement for the morphological evolution of cracks. However, one of the exemptions would be hydraulic fracturing where the crack propagation is driven by the fluid pressure which strongly depends on the crack opening displacement. In this study, we look into two known approaches, a line integral and a level-set method, for crack opening computation mainly from an implementation point of view. Firstly, we derive an approximation of a discontinuous function field in the variational phase-field setting which is then applied to obtain the crack opening (displacement jump) and verify the approximation against a closed solution. We then propose a “certain distance from the crack” required for the level-set function using a one-dimensional analysis. Finally, we compare these approaches under several different conditions such as crack alignment to the mesh or under loading (asymmetric displacement field) and investigate the convergence with respect to the mesh size.

Fracture of hyperelastic materials such as synthetic rubber, hydrogels, textile fabrics is an essential problem in many engineering fields. The computational simulation of such a fracture is complicated, but the use of phase field models (PFMs) is promising. Indeed, in PFMs, sharp cracks are not treated as discontinuities; instead, they are approximated as thin damage bands. Thus, PFMs can seamlessly model complex crack patterns like branching, merging, and fragmentation. However, previous PFMs for hyperelastic materials, which are mostly based on a PFM with a simple quadratic degradation function without any user-defined parameters, provide solutions that are sensitive to a length scale (that controls the width of the damage band). The current practice of considering this length scale as a material parameter suffers from two issues. First, such a calculated length scale might be too big (compared with the problem dimension) to provide meaningful crack patterns. Second, it might be too small, which results in undesirable computationally expensive simulations. This paper presents a length scale insensitive PFM for brittle fracture of hyperelastic materials. This model is an extension of the model of Wu (2017) with a material parameter dependent rational degradation function, which converges to Cohesive Zone Model (CZM) at least for 1D problems (Wu and Nguyen, 2018), and also can deal with crack nucleation and propagation simultaneously. Results of mode-I and mixed-mode fracture problems obtained with the method of finite elements are in good agreement with previous findings and independent of the discretization resolution. Most importantly, they are independent of the incorporated length scale parameter. Moreover, preliminary results show that the proposed model is as efficient as, if not more than the previous models.

In this work, we propose a thermodynamically consistent phase-field model for the brittle fracture analysis of thick plates. A hybrid model, which is fast and accurate, is proposed for the phase-field modeling of fracture in thick plates. Reddy’s third-order shear deformation theory (TSDT) has been employed to capture the transverse shear deformation effects in thick plates. Governing equations are derived by seeking the minimization of the free-energy functional. A staggered solution algorithm with arc length control is used to solve the governing equations within the finite element framework. The nucleation and propagation of cracks in the thick plates subjected to uniformly distributed load is presented. The mechanical response corresponding to phase-field models based on both the classical plate theory and TSDT has been compared for the case of thick plates and a significant difference between these two models is observed. Parametric studies have been carried out to illustrate the effects of boundary conditions, shear deformation, and the mesh size.

A thermal fluctuation-based nucleation model for use in a phase-field model is proposed. In the model, a normally distributed random temperature fluctuation is imposed on a calculation system, and if a given grid temperature is lower than the average, the undercooling is converted to an increase in the crystallized/precipitated phase fraction. The conversion coefficient is derived as a function of a phase-field critical value for nucleation. The idea of a threshold of nucleation for the phase field, i.e., a critical value, is introduced first in the present study. The model is applied to homogeneous nucleation during solidification in a supercooled pure metal, and a reasonable nucleation probability curve is successfully obtained from the numerical calculations. An application to the homogeneous nucleation of an Al-Cu alloy is also demonstrated.

The predictive description of fracture has been an important topic of research over the last decades. In this context, the recently developed phase-field approach to fracture has proven to be an efficient and versatile tool in handling arbitrary and topologically complex crack patterns in an adequate manner. However, open problems reside in the distinction between tensile and compressive states in the framework of energetically driven crack evolution, especially at finite deformations. In the present contribution, a novel methodology to split energy-density functions is proposed that can be applied to a large variety of polyconvex energies. The applicability of the proposed formulation is validated by a series of numerical examples involving small and large deformations, in which we also provide comparisons with existing splits from literature.