Three-dimensional Phase-field Model Research Articles

Modeling the dynamic and quasi-static compression-shear failure of brittle materials by explicit phase field method

The phase field method is a very effective method to simulate arbitrary crack propagation, branching, convergence and complex crack networks. However, most of the current phase-field models mainly focus on tensile fracture problems, which is not suitable for rock-like materials subjected to compression and shear loads. In this paper, we derive the driving force of phase field evolution based on Mohr–Coulomb criterion for rock and other materials with shear frictional characteristics and develop a three-dimensional explicit parallel phase field model. In spatial integration, the standard finite element method is used to discretize the displacement field and the phase field. For the time update, the explicit central difference scheme and the forward difference scheme are used to discretize the displacement field and the phase field respectively. These time integration methods are implemented in parallel, which can tackle the problem of the low computational efficiency of the phase field method to a certain extent. Then, three typical benchmark examples of dynamic crack propagation and branching are given to verify the correctness and efficiency of the explicit phase field model. At last, the failure processes of rock-like materials under quasi-static compression load are studied. The simulation results can well capture the compression-shear failure mode of rock-like materials.

Phase field method for simulating the brittle fracture of fiber reinforced composites

We implemented a three-dimensional phase field model to simulate crack initiation and propagation in fiber reinforced composites. An efficient and robust staggered scheme is adopted by decoupling the fracturing phase field and displacement field. With this approach, the mixed-mode crack propagation in composite materials as well as the fracture process of fiber reinforced composites can be precisely captured. Our results revealed that the mechanical properties and crack propagation are influenced by the direction of inside imperfection with respect to load, interactive effects of the imperfections and damageable layer impact. We observed that accurate post-peak displacements and sharper crack evolution can be obtained by employing smaller displacement increments. The present model precisely captured the physical-consistent phenomena that crack initiates from a damageable layer, followed by matrix cracking, crack branches when it encounters a stiffer material and propagates along the interface, and interactive crack growth.

Phase field simulation to one-way shape memory effect of NiTi shape memory alloy single crystal

Based on the Ginzburg-Landau theory and in the framework of thermodynamics, a three-dimensional phase field model was developed to simulate the one-way shape memory effect (OWSME) of NiTi shape memory alloy (SMA) single crystal originated from the martensite transformation and reorientation. A new reorientation strain was simultaneously introduced when a transformation strain was considered in the proposed model, and the phase field equations corresponding to the martensite transformation and reorientation were derived, respectively. From the simulations to the OWSME, it is found that: the martensite variants are grown in a self-coordinating manner and combined by twinned interfaces to reduce the local elastic energy during the temperature-induced transformation from B2 phase to B19′ one; then, the stress-induced martensite reorientation is completed through the migration of twinned interfaces and the variants with optimal orientations are obtained during the subsequent loading, and a residual strain is produced after the unloading; during the further heating after the unloading, a uniform temperature-induced reverse transformation from the single-oriented martensite variant to B2 austenite phase occurs, and the residual strain produced by the stress-induced martensite reorientation is progressively recovered.

Three-dimensional phase field modeling of inhomogeneous gas-liquid systems using the PeTS equation of state.

Recently, an equation of state (EoS) for the Lennard-Jones truncated and shifted (LJTS) fluid has become available. As it describes metastable and unstable states well, it is suited for predicting density profiles in vapor-liquid interfaces in combination with density gradient theory (DGT). DGT is usually applied to describe interfaces in Cartesian one-dimensional scenarios. In the present work, the perturbed LJ truncated and shifted (PeTS) EoS is implemented into a three-dimensional phase field (PF) model which can be used for studying inhomogeneous gas-liquid systems in a more general way. The results are compared with the results from molecular dynamics simulations for the LJTS fluid that are carried out in the present work and good agreement is observed. The PF model can therefore be used to overcome the scale limit of molecular simulations. A finite element approach is applied for the implementation of the PF model. This requires the first and second derivatives of the PeTS EoS which are calculated using hyper-dual numbers. Several tests and examples of applications of the new PeTS PF model are discussed.

Phase-field modeling of θ′ precipitation kinetics in 319 aluminum alloys

Understanding the morphological evolution of precipitates is critical for evaluating their hardening effects and therefore improving the yield strength of an alloy during aging. Here we present a three-dimensional phase-field model for capturing both the nucleation and the growth kinetics of the precipitates and apply it to modeling θ′ precipitates in 319 aluminum alloys. The model incorporates the relevant thermodynamic data, diffusion coefficients, and the anisotropic misfit strain from literature, together with the anisotropic interfacial energy from first-principles calculations. The modified classical nucleation theory is implemented to capture the nucleation kinetics. The model parameters are optimized by comparing the simulation results to the experimentally measured peak number density, average diameters, average thicknesses and volume fractions of precipitates during isothermal aging at 463 K (190 °C), 503 K (230 °C) and 533 K (260 °C). Further model improvements in terms of prediction accuracy of the precipitate kinetics in 319 alloys and the remaining challenges are discussed.

Three-dimensional phase field simulation of intragranular void formation and thermal conductivity in irradiated α-Fe

Nucleation and growth of the irradiation-induced void, which is a fatal concern for the thermal conductivity of a structural material, is highly sensitive to the extreme irradiation conditions. A three-dimensional phase field model in conjunction with an explicit nucleation algorithm has been developed and utilized to investigate this issue. The temperature-dependent material parameters of α-Fe are derived from ab initio calculations. Phase field simulations reveal the effects of temperature and damage rate on the void formation behavior. At the same temperature, lower damage rate leads to longer incubation time, smaller void radius and less void number, while at the same damage rate, higher temperature causes shorter incubation time, larger void radius but less void number. The effective thermal conductivity of irradiated α-Fe is examined and analyzed by taking into account of void volume fraction, which agrees well with the calculated results using the Maxwell and Bauer models. The void formation behavior revealed by the phase field simulations and the effect of void volume fraction on thermal conductivity could be helpful in understanding the irradiation damage of structural materials.

Effect of applied strain on phase separation of Fe–28 at.% Cr alloy: 3D phase-field simulation

A quantitative simulation of the separation of the α′ phase in Fe–28 at.% Cr alloy under the effects of applied strain is performed by utilizing a three-dimensional phase-field model. The elongation of the Cr-enriched α′ phase becomes obvious with the influence of applied uniaxial strain for the phase separation transforms from spinodal decomposition of 700 K to nucleation and growth of 773 K. The applied strain shows a significant influence on the early stage phase separation, and the influence is enlarged with the elevated temperature. The steady-state coarsening with the mechanism of spinodal decomposition is substantially affected by the applied strain for low-temperature aging, while the influence is reduced as the temperature increases and as the phase separation mechanism changes to nucleation and growth. The peak value of particle size distribution decreases, and the PSD for 773 K becomes more widely influenced by the applied strain. The simulation results of separation of the Cr-enriched α′ phase with the applied strain provide a further understanding of the strain effect on the phase separation of Fe–Cr alloys from the metastable region to spinodal regions.

Multi-GPU hybrid programming accelerated three-dimensional phase-field model in binary alloy

In the process of dendritic growth simulation, the computational efficiency and the problem scales have extremely important influence on simulation efficiency of three-dimensional phase-field model. Thus, seeking for high performance calculation method to improve the computational efficiency and to expand the problem scales has a great significance to the research of microstructure of the material. A high performance calculation method based on MPI+CUDA hybrid programming model is introduced. Multi-GPU is used to implement quantitative numerical simulations of three-dimensional phase-field model in binary alloy under the condition of multi-physical processes coupling. The acceleration effect of different GPU nodes on different calculation scales is explored. On the foundation of multi-GPU calculation model that has been introduced, two optimization schemes, Non-blocking communication optimization and overlap of MPI and GPU computing optimization, are proposed. The results of two optimization schemes and basic multi-GPU model are compared. The calculation results show that the use of multi-GPU calculation model can improve the computational efficiency of three-dimensional phase-field obviously, which is 13 times to single GPU, and the problem scales have been expanded to 8193. The feasibility of two optimization schemes is shown, and the overlap of MPI and GPU computing optimization has better performance, which is 1.7 times to basic multi-GPU model, when 21 GPUs are used.

Three-dimensional phase-field simulations of intragranular gas bubble evolution in irradiated U-Mo fuel

The evolution of fission gas bubbles in irradiated materials plays a critical role in the microstructural processes that leads to dimensional changes of U-Mo alloy fuels, e.g., fuel swelling. Although the intergranular bubbles-induced fuel swelling has been long-discussed for U-Mo fuel, there are very few computational studies of the formation of intragranular gas bubbles and its impact on fuel swelling. To this end, we develop a three-dimensional phase-field model to investigate the evolution of intragranular gas bubbles in U-Mo fuel. Fission induced defect formation and annihilation processes, such as vacancy-interstitial recombination, fission gas atom resolution, and interactions with dislocations and grain boundaries are incorporated in the model. Simulations show that the intragranular gas bubbles can be stabilized to certain sizes due to the balance between the generation and annihilation of defects. The intragranular gas bubbles induced fuel swelling is predicted to be comparable to experimental measurements. The effects of the irradiation and fuel fabrication conditions (i.e., fission rate, fuel grain size, and mechanical work induced deformation) on the bubble evolution and the resultant swelling are investigated. The current simulations provide a better understanding of intragranular gas bubble-induced swelling and a solid foundation for the future study of the nucleation and growth of intergranular gas bubbles and recrystallization in U-Mo fuel.

Phase field modeling for cyclic phase transition of NiTi shape memory alloy single crystal with super-elasticity

From the Ginzburg-Landau’s theory and thermodynamics, a three-dimensional phase field model was developed to simulate the cyclic phase transition of the NiTi shape memory alloy (SMA) single crystal with super-elasticity. The phase transition from the austenite phase to martensite one and its reverse were numerically simulated by the developed phase field model considering the repeated tension-unloading. The simulations indicated the nucleation and evolution of the multi-variant martensite and austenite phases in a scale of single crystal during the cyclic phase transition. It is found that the NiTi SMA single crystal with super-elasticity exhibits a remarkable localized deformation, accompanied by a local softening feature. Moreover, the microstructure evolution of the NiTi single crystal during the cyclic deformation was investigated. It is shown that the super-elasticity of the NiTi single crystal degrades gradually until it reaches to a stable state after certain load cycles, and a progressive accumulation of remnant martensite phase occurs, simultaneously; the phase transition of the NiTi single crystal gradually alters from a local mode into a uniform one, which comes from the internal defects and their evolutions; a large number of remnant martensite phase occurs because the enriched defects can greatly prevent the occurrence of reverse phase transition.

Large-Aperture and Grain-Boundary Engineering through Template-Assisted Metal Dewetting for Resonances in the Short Wave Infrared

We extend the fabrication method of template-assisted metal dewetting (TeAMeD) to create near-infrared resonant nanostructures in an Au film without the need for etching or lift-off. TeAMeD has previously been used to generate high aspect-ratio sub-10 nm apertures, but struggles to generate larger apertures (>100 nm). In this work, we introduce a method to create larger apertures using templates consisting of fin-like patterns with radial symmetry. We also report evidence of grain boundary engineering, through the template pinning effect. Our three-dimensional phase field model of TeAMeD predicts both the grain-boundary pinning and aperture opening effects that agree well with experiments. Combined with simulation design, TeAMeD can be established as a grain engineering platform, allowing grain shape and boundary position to be controlled. Variations of template motif produce larger grains and numerous possible outcomes, including suspended Au nanodisks and triangular apertures.

Growth kinetics and morphology of snowflakes in supersaturated atmosphere using a three-dimensional phase-field model.

Simulating ice crystal growth is a major issue for meteorology and aircraft safety. Yet, very few models currently succeed in reproducing correctly the diversity of snow crystal forms, and link the model parameters to thermodynamic quantities. Here, we demonstrate that the new three-dimensional phase-field model developed in Demange etal. [npj Comput. Mater. 3, 1 (2017)2057-396010.1038/s41524-017-0015-1] is capable of reproducing properly the morphology and growth kinetics of snowflakes in supersaturated atmosphere. Aside from that, we show that the growth dynamics of snow crystals satisfies the selection theory, consistently with previous experimental observations. Finally, we link the parameters of the phase-field model to atmospheric parameters.

Experimental investigations and phase-field simulations of triple-phase-separation kinetics within liquid ternary Co-Cu-Pb immiscible alloys.

The phase-separation kinetics and microstructure evolution mechanisms of liquid ternary Co_{43}Cu_{40}Pb_{17} immiscible alloys are investigated by both the drop tube technique and phase-field method. Two successive phase separations take place during droplet falling and lead to the formation of a three-phase three-layer core-shell structure composed of a Co-rich core, a Cu-rich middle layer, and a Pb-rich shell. The Pb-rich shell becomes more and more conspicuous as droplet diameter decreases. Meanwhile, the Co-rich core center gradually moves away from the core-shell center. Theoretical analyses show that a larger temperature gradient inside a smaller alloy droplet induces the accelerated growth of the surface segregation shell during triple-phase separation. The residual Stokes motion and the asymmetric Marangoni convection result in the appearance of an eccentric Co-rich core and the core deviation degree is closely related to the droplet size and initial velocity. A three-dimensional phase-field model of ternary immiscible alloys, which considers the successive phase separations under the combined effects of Marangoni convection and surface segregation, is proposed to explore the formation mechanisms of three-phase core-shell structures. The simulated core-shell morphologies are consistent with the experimental observations, which verifies the model's validity in reproducing the core-shell dynamic evolution. Numerical results reveal that the development of three-phase three-layer core-shell structures can be attributed to the primary and then secondary phase separations dominated simultaneously by Marangoni convection and surface segregation. Furthermore, the effects of droplet temperature gradient on the growth kinetics of the surface segregation shell are analyzed in the light of phase-field theory.

Three-dimensional phase-field model of dislocations for a heterogeneous face-centered cubic crystal

A central aim of current materials studies is to develop a predictive modeling that incorporates dislocation-based plastic activity and microstructural evolution. Phase-field method has emerged as a powerful tool for addressing this issue, providing us with a versatile variational framework able to describe the movement of dislocations in interaction with underlying microstructures. In this article, a three-dimensional phase-field model of dislocations (PFMD) is developed with a discretization scheme that explicitly captures the face-centered cubic (FCC) geometry. Within this framework, continuous fields are discretized in a way that allows to consider strongly heterogeneous materials and sharp interfaces (free surfaces, stiffer precipitates, pores...) without generating numerical artifacts. The PFMD exposed in this work reproduces dislocation activity in FCC geometry, their reactions, and a particular attention is devoted to the dislocation core behaviors in order to remove effects present in prior generic PFMDs that can appear to be spurious for micron-scale applications. This allows us to rigorously reproduce the dislocation’s velocity with respect to experimental friction coefficients. The model is discussed and illustrated by applications standing at different space-scales that show how dislocations operate with microstructural heterogeneities such as free-surfaces (cylindrical nanopillar) and voids (pore under isostatic pressure).

Phase-field modeling of void anisotropic growth behavior in irradiated zirconium

A three-dimensional phase-field model was developed to study the effects of surface energy and diffusivity anisotropy on void growth behavior in irradiated Zr. The gamma surface energy function used in the phase-field model was developed with the surface energy anisotropy calculated from the molecular dynamics simulations. The model assumes that vacancies have larger mobility in the c-axis than in the a-axis and b-axis, whereas interstitials have larger mobility in the basal plane than in the c-axis. The equilibrium void morphology and the effect of defect concentrations and mobility anisotropy on void growth behavior were simulated using the developed model. The simulations demonstrated that (1) the developed phase-field model correctly reproduces the faceted void morphology predicted by the Wulff construction; (2) with isotropic diffusivity, the void prefers to grow on the basal plane; and (3) when the vacancy has large mobility along the c-axis and interstitial has a large mobility on the basal plane of hexagonal closed-packed Zr alloys, a platelet void grows in the c-direction and shrinks on the basal plane. These findings are in agreement with the observation of void morphology and growth behavior in irradiated Zr, and reveal that the strong mobility anisotropy of defects results in the anisotropic growth of voids in Zr.

Self-Assembly of Islands on Spherical Substrates by Surface Instability.

Through strain-induced morphological instability, protruding patterns of roughly commensurate nanostructures are self-assembled on the surface of spherical core/shell systems. A three-dimensional (3D) phase field model is established for a closed substrate. We investigate both numerically and analytically the kinetics of the morphological evolution, from grooves to separated islands, which are sensitive to substrate curvature, misfit strain, and modulus ratio between the core and shell. The faster growth rate of surface undulation is associated with the core/shell system of a harder substrate, larger radius, or misfit strain. On the basis of a Ag core/SiO2 shell system, the self-assemblies of SiO2 nanoislands were explored experimentally. The numerical and experimental studies herein could guide the fabrication of ordered quantum structures via surface instability on closed and curved substrates.

Understanding the facet formation mechanisms of Si thin-film solidification through three-dimensional phase-field modeling

Adaptive phase field modeling is used in order to model the formation mechanism of a silicon faceted interface in three dimensions. We investigate the faceting condition for equilibrium shapes and dynamic situations. In this study, we propose a new anisotropic function of surface energy for the phase-field simulations in three-dimension, and negative stiffness is further considered. The morphological evolutions are presented and compare well with experimental findings. The growth mechanism is further discussed.

A phase-field method coupled with CALPHAD for the simulation of ordered [formula omitted]-carbide precipitates in both disordered [formula omitted] and [formula omitted] phases in low density steel

In order to simulate multi-component diffusion controlled precipitation of ordered phases in low density steels using the phase-field method, the Gibbs free energy of the γ,α and κ phases in the quaternary Fe-Mn-Al-C system was linked to the CALPHAD method using a three-sublattice model which is based on the accumulation of considerable thermodynamic data in multi-component systems and the assurance of continuous variation of the interface area. This model includes the coherent precipitation of κ phase from a disordered FCC γ phase and semi-coherent precipitation of the same κ phase from a disordered BCC α structure. The microstructure evolution of κ-carbide was simulated with three-dimensional phase-field model. The simulation was first performed for a single particle in both γ and α phases to investigate the evolution of interfacial and elastic strain energy during the precipitation process. The simulation results show that κ has a cuboidal morphology in γ and elongated plate-like morphology in α which is in agreement with the morphologies reported in the literature. The multi-particle simulations were also performed for the precipitation of κ phase from both disordered γ and α. The results also demonstrate that the size of κ precipitates in γ is remarkably smaller than that in α phase.

Analysis for wetting on rough surfaces by a three-dimensional phase field model

In this paper, we consider the derivation of the modified Wenzel's and Cassie's equations for wetting phenomena on rough surfaces from a three-dimensional phase field model. We derive an effective boundary condition by asymptotic two-scale homogenization technique whenthe size of the roughness is small. The modified Wenzel's and Cassie'sequations for the apparent contact angles on the rough surfacesare then derived from the effective boundary condition. Thehomogenization results are proved rigorously by the$\Gamma$-convergence theory.

Interface stress evolution of martensitic transformation in MnCu alloys: A phase-field study

Martensitic transformations (MTs) in shape memory alloys (SMAs) usually give rise to internal elastic stress, because of the shape change from austenite to martensite. The stress condition of the interface plays a key role in the thermoelastic transformation behaviour and shape memory effect. In order to investigate the stress field originated from MT, a three-dimensional (3D) phase-field model was applied to the face-centered cubic structure to face-centered tetragonal structure (fcc-to-fct) MT in a MnCu alloy. The following results were obtained: The interface stresses on the invariant habit plane arise with the thickening of martensite plate; When the habit plane is not an invariant plane, the stress state around the interface changes abruptly; The formation of transformation twin is resulted from the fluctuant interface stress, the macroscopic habit plane has not been well obtained, and the elastic strain energy increases with the growth of twinning martensite; Martensitic block containing several twinning domains can be generated from a single-variant embryo, and the fluctuant elastic stresses have retained in the block.