Three-dimensional Phase-field Model Research Articles

Three-dimensional phase field modeling, orientation prediction and stress field analyses of twin-twin, twin-grain boundary reactions mediated by disclinations in hexagonal close-packed metals

Crystalline defects, such as dislocations, disclinations, twins and grain boundaries, play critical roles in determining the mechanical properties of metals and alloys. In particular, with multiple competitive deformation modes activated, the mechanical behaviors of hexagonal close-packed metals are strongly influenced by the interactions and reactions of various types of defects. Despite extensive studies on the elastic interactions of defects, a theoretical framework capturing crystallographic reactions, especially reaction products and associated local stress concentration, is still unavailable. Here we suggest a disclination-based method to quantify defect reactions. By using a combination of crystallographic calculations and phase field modeling/simulations, twin-twin and twin-grain boundary reactions in hexagonal close-packed metals have been quantitatively analyzed. It has been found that partial disclinations, accompanied with other defects (e.g., {112¯6} and {112¯2} high-index twins), can be generated by defect reactions as typical byproducts. The orientation change and stress fields caused by disclination formation have been systematically calculated, which offers a rigorous mathematical foundation to explore twin-twin, twin-grain boundary reactions. By quantitatively determining defect reactions and local stress fields, our work provides new insights into the deformation mechanism and microstructure-property relationship in metallic materials.

A three-dimensional coupled thermo-elastic-plastic phase field model for the brittle-ductile failure mode transition of metals

Dynamic brittle fracture and shear banding are two typical failure modes of metals, and the transformation of the brittle-ductile failure mode has been observed in the Kalthoff test. This paper establishes a thermo-elastic-plastic coupled three-dimensional phase field model to simulate brittle-ductile failure mode transition of metals. The expression for the variation of the Taylor-Quinney coefficient with stress triaxiality is adopted, and the critical energy release rate is automatically adjusted using the Taylor-Quinney coefficient. Then, the Kalthoff test is simulated using the proposed model. The brittle-ductile failure mode transformation phenomenon is reproduced, which agrees well with the experimental results. It can be well proved that impact velocity is crucial in determining the transition to failure mode. At low-velocity impact, the energy is insufficient to drive the plastic accumulation of the shear band, resulting in brittle tensile fracture. At high-velocity impact, the energy is sufficient to drive the formation of adiabatic shear bands, resulting in tensile shear failure. In addition, three-dimensional simulations show that the tip of the shear band exhibits a crescent-shaped non-two-dimensional extension state under finite thickness. This numerical framework provides a predictive tool to understand the evolution of the dynamic failure of metals under impact loading.

A three-dimensional phase-field model for studying the orientation-dependent interface evolution in stress-induced martensitic phase transformation

In this study, we address the challenging task of predicting the motion of interfaces in stress-driven martensitic phase transformations within shape memory alloys. The novelty of our approach lies in the introduction of a monolithically solved thermodynamic-based phase-field method, specifically tailored for large strain conditions at the nanoscale for three-dimensional problems. To achieve this, we have developed a custom finite element software seamlessly integrated into the FEniCS open-source framework, providing a sophisticated and efficient tool for the examination of nanostructure evolution. Our investigation delves into five distinct and complex scenarios under diverse loading conditions for two– and three-dimensional problems, each presenting unique challenges: (i) a straightforward uniaxial tension scenario featuring a square domain with a pre-existing martensitic nucleus; (ii) the evolution of interfaces in a square sample incorporating a circular central nanovoid under biaxial tensile stress; (iii) the analysis of a rectangular beam subjected to horizontal and vertical compressive loads; (iv) the assessment of an initially voided rectangular beam experiencing mixed loading conditions; and (v) the three-dimensional simulations of cubic-to-tetragonal phase transformation using various orientation of the habit plane. In contrast to prior studies, our analysis not only explores the standard factors of habit plane reorientation and strain transformation values but also delves into the impact of nanovoids on austenite–martensite interface evolution.

Effect of solutes on texture evolution during grain growth in ZK60 alloy by phase field simulation

A three-dimensional (3D) multiscale phase field model was proposed to investigate the effect of Ca, Y and Al solutes on the texture evolution of ZK60 alloy during grain growth at 573 K under an applied compression stress of 270 MPa, in which the characteristics of the elastic anisotropy and grain boundary (GB) segregation caused by the solutes were taken into account. The results show that either Ca or Y addition can weaken the basal texture of ZK60, while Al addition enhances it. The deviation of the grain size distribution from the Hillert model becomes greater with the addition of Al, while Ca or Y addition reduces the deviation. The solute segregation-induced GB energy change has a very limited effect on the texture evolution compared with elastic anisotropy. The revealed mechanism is of great significance for the microstructure design of the ZK60 alloy by an appropriate alloying strategy.

Elastoplastic damage analysis and structural optimization of soluble bridge plug based on phase field method

Bridge plugs play a crucial role as a plugging tool for downhole staging in the staged fracturing process. In particular to the soluble bridge plug, which can be automatically dissolve in that backflow fluid after hydraulic fracture. In this paper, a single silp with small diameter soluble bridge plug is designed to take into account its passability in the casing deformation well. A three-dimensional phase field model of the elastoplastic anchoring mechanism was developed to investigate the maximum pressure-bearing of the designed soluble bridge plug during the fracturing and optimize its structure. The phase field model we use is suitable for linear isotropic hardening materials, in which the elastic energy is decomposed into compression and tension parts, and the plastic energy is decomposed into plastic free energy and plastic dissipation energy. The model is assumed to be quasi-static. Tensile experiments with two asymmetrically notched metal bars are used to verify the applicability of our model to materials with small plastic strains. The simulation shows that the designed bridge plug can adapt to the working pressure differential of 70 MPa, and the optimal number of slip teeth is 8 or 9.

Three-dimensional phase-field simulation of intergranular bubble evolution in UO2 during irradiation

In this study, a three-dimensional phase-field model was developed to investigate the evolution of intergranular bubbles during irradiation. The model considered several grain structures and multiple fission gas bubbles, with the microstructure evolution being governed by improved dynamic equations. Verification of the phase-field model showed good agreement with the theory in describing the properties of grain boundary (GB). The study examined the dependency of bubble percolation on bubble shape, fission rate, and average grain size. The simulation results revealed that the changes in these three factors have significant effects on the shape of GB bubbles, the percolation rate, and the connectivity threshold of GB bubbles. The bubble connection threshold was obtained from the curve of triple junction coverage versus grain boundary coverage. The results indicate that the connectivity threshold increases with the higher fission rate and lager average grain size. This investigation into bubble percolation during irradiation provides valuable insights for further study on fission gas release in UO2 fuel within nuclear reactors.

Effect of concurrent thermal grooving and grain growth on morphological and topological evolution of a polycrystalline thin film: Insights from a 3D phase-field study

Experiments and computer simulations of bicrystals have been extensively used to study the role of thermal grooving on grain boundary motion. Although these studies provide a fundamental understanding of the factors that affect boundary motion, they do not capture interactions between the grain boundary network and (solid-vapor) surface on the overall grain growth kinetics. To understand the complex interactions between the thermal groove and the grain boundary, we developed a three-dimensional phase-field model to simulate concurrent capillary-driven grain growth and surface diffusion-controlled thermal grooving. We systematically studied the role of film thickness and surface diffusivity on the interactions between thermal grooving and grain growth in polycrystalline films. To make outcomes statistically significant, we implement an efficient active parameter tracking algorithm that can simulate a large number of grain orientations. Our results show an increase in the degree of stagnation with the reduction in film thickness due to the modification of boundary curvature normal to the surface of the film. Large-scale simulations of polycrystalline films show the development of nonuniform groove surface profiles with varying degrees of asymmetry that correspond to steady state, decelerating, and stationary boundaries. The stagnation of grain boundaries shows dependence on groove depth and topological distribution of grains. The critical grain size for grain boundary stagnation and grain size distribution varies with film thickness. We also demonstrate, using various tailored configurations, how concomitant thermal grooving can lead to the complete arrest of grain growth.

Improving long-term solid oxide fuel cell stack performance prediction accuracy using a microstructure evolution model

Solid oxide fuel cells (SOFCs) are highly efficient energy conversion devices capable of utilizing a wide range of fuels. However, a variety of degradation phenomena affects their long-term performance and durability. The purpose of this study is to explain an increase in voltage of an SOFC stack observed during a 3800 h-long period of operation. A three-dimensional phase field model using focused ion beam scanning electron microscopy data as initial input is used to simulate the evolution of an SOFC microstructure over a period of prolonged operation. For the generated time steps, microstructure parameters are obtained, and a microstructure-scale mass and charge transport model is used to predict the voltage change. The microstructure evolution model, combined with the electrochemical performance model, predicts the 0.01 V increase in terminal voltage during the 3800-long period of operation, which is consistent with the experimental results. The behavior is predicted for various phase mobility ratios, suggesting that the phenomenon is related to the specific characteristics of the particular microstructure investigated in the study. Clearing of diffusion pathways is identified as the root cause of the performance enhancement. Additionally, the results suggest non-negligible changes in the ceramic phase within the electrode.

Three-dimensional experimental-scale phase-field modeling of dendrite formation in rechargeable lithium-metal batteries

This paper presents a phase-field based numerical study on the 3D formation of dendrites due to electrodeposition in an experimental-scale lithium metal battery. Small-scale 3D simulations were firstly conducted to elucidate the characteristics and resolution requirements of the numerical framework. Using a four-fold anisotropy model to simulate the growth of lithium deposition, the dependency of dendrite morphology on charging conditions (ϕb=−0.7V and ϕb=−1.4V) on a (larger) experimental-scale metal anode was demonstrated. The dendrite shape was found to shift from a smoother, tree-like formation at the lower applied voltage, to a more spike-like, highly branched structure at the higher voltage. The resulting morphological parameters, such as dendrite propagation rates, volume-specific area, and side branching rates, were compared against published experimental data and found to be comparable to the reported ranges for the electrodeposition of spike- or tree-like metal dendrites. This finding supports our previous observation that dendrite formation is connected to the competition between the lithium cation diffusion and electric migration, generating an uneven distribution of Li+ on the electrode surface. This observation also gives insight into dendrite inhibition strategies focusing on enhancing the diffusion of lithium ions to achieve a more uniform concentration field on the anode surface.

Effect of surface energy anisotropy on hole growth in a single-crystalline thin film: A phase-field study

Prediction of the temporal evolution of a pre-existing hole on a solid-state thin film has important implications for self-organized nanoscale pattern formation. In this work, we employ a three-dimensional phase-field model to elucidate the mechanism of corner instability during capillary-driven hole growth in a single-crystalline free-standing thin film. We perform a systematic study to demonstrate the effect of crystalline anisotropy in surface energy, surface diffusivity, and film thickness on hole growth. Our simulations reveal that the instability during hole growth arises due to saturation of the rim height at the corner of the hole that is directly related to the arc length of the corner, specified by the anisotropy in surface energy. Our investigation further reveals the presence of a time-invariant shape of the corner with the onset of corner instability.

A new three dimensional cumulant phase field lattice Boltzmann method to study soluble surfactant

Surfactants play a critical role in the physics of paint and coating formulations, affecting key rheological properties such as viscosity, yield stress, and thixotropy. This paper proposes a new three-dimensional phase-field model that uses the cumulant lattice Boltzmann method (LBM) to simulate soluble surfactants. Although current phase-field models commonly use Langmuir's relationship, they cannot calculate interfacial tension analytically, or the LBM models used are unstable when viscosities are low. However, the proposed method overcomes these limitations through two main features. First, the main parameters for modeling and controlling the surfactant's strength and interaction with other phases are directly obtained from a given initial interfacial tension and bulk surfactant, eliminating the need for trial-and-error simulations. Second, a new equilibrium distribution function in the moment space that includes diagonal and off diagonal elements of the pressure tensor is used to minimize Galilean invariance violation. Additionally, there is no need to use an external force to recover multiphase flows, which could break mass conservation. Furthermore, this method has significant potential for parallelization since only one neighbor's cell is used for discretization. The method shows Langmuir relation behavior and is validated with analytical solutions for various interfacial tensions and surfactant concentrations. Moreover, the paper demonstrates the influence of interfacial tension and surfactants on spurious velocities, indicating the method's stability at low viscosities. The dynamics of droplets in the presence of the surfactants is studied in spinodal decomposition and under various external forces. The method accurately simulates the breaking-up and coalescence for these cases. Furthermore, the method successfully simulates the breakage of a liquid thread at a high viscosity ratio.

Generation mechanism and motion behavior of sliver defect in single crystal Ni-based superalloy

Sliver is a common but easily neglected defect in single crystal Ni-based superalloy castings. To date, there is still no unified viewpoint on its formation mechanism and generation causes. In this work, the orientation discontinuity and motion behavior of sliver defects were studied through experiments and numerical simulations. The ultrathin wedge-shaped specimen containing the grain boundary of the sliver and the matrix was prepared at the initial position of the sliver defect for the observation of equal thickness fringes. The discontinuity of equal thickness fringes on both sides of the grain boundary was observed through a transmission electron microscope, which directly confirms the abrupt change in the orientation between the sliver and matrix from the nanoscale. The crystal lattices at the smooth area and the bulging area of the grain boundary were found to have unusually different arrangements. The irregular lattice arrangement at the bulging area shows that the grain boundary has experienced high-stress deformation. Statistical results of sliver orientation deviation with a further composition analysis show the micro protuberance of the mold shell has a noticeable inductive effect on the sliver generation. Furthermore, a self-developed three-dimensional phase-field simulation model coupled with the spatial topology algorithm is established to simulate the orientation deflection behavior and orientation deviation threshold of fractured dendrites. The simulation results indicated that there is an upper limit of the cross-section solid fraction at the fracture position for the motion of the fractured dendrites. When the cross-section solid fraction at the fracture position is higher than this upper limit, it will be difficult to produce large deviation slivers due to the structural limitation of surrounding dendrites. This upper limit does not change with the solidification temperature gradient.

Phase-field-method-studied mechanism of Cu-rich phase precipitation in Al<sub><i>x</i></sub>CuMnNiFe high-entropy alloy

High-entropy alloys with BCC and FCC coexisting structures usually have excellent comprehensive mechanical properties, and Al element can promote the transformation of Cu-containing high-entropy alloys from FCC structure to BCC structure to obtain the BCC and FCC coexisting structures. In order to illustrate the process of phase separation of high entropy alloys, a low-cost Al-TM transition group element high-entropy alloy is selected in this work. Based on the Chan-Hilliard equation and Allen-Cahn equation, a three-dimensional phase field model of Al<sub><i>x</i></sub>CuMnNiFe high-entropy alloy is established, and the microscopic evolution of the nano-Cu-rich phase of Al<sub><i>x</i></sub>CuMnNiFe high-entropy alloy (<i>x</i> = 0.4, 0.5, 0.6, 0.7) at 823 K isothermal aging is simulated. The results show that the Al<i><sub>x</sub></i>CuMnNiFe high-entropy alloy generates two complex core-shell structures upon aging: Cu-rich core/B2<sub>s</sub> shell and B2<sub>c</sub> core/FeMn shell, and it is found through discussion and analysis that the formed B2<sub>c</sub> plays an inhibitory role in the formation of the nano-Cu-rich phase, and that this inhibitory role becomes larger with the increase of Al element. Combining the empirical formula, the curve of yield strength of the Cu-rich phase varying with the aging time is obtained for the Al<sub><i>x</i></sub>CuMnNiFe high-entropy alloy, and the overall yield strength of the high-entropy alloy has a rising-and-then-falling trend with the change of time, and the aging time of the peak yield strength and the alloy system are obtained from the change of the curve, so that the best alloy system and aging time of the high-entropy alloy can provide a reference for aging process.

Phase-field method based simulation of martensitic transformation in porous alloys

Porous materials, characterized by the presence of interconnected pores, exhibit the properties different from their bulk counterparts. One of properties of interest is that the pores can influence the martensitic transformation in shape memory alloys (SMAs), which directly affects the material's shape memory effect and mechanical properties. The martensitic transformation is accompanied by the formation of different martensitic variants, which determine the overall morphology, distribution, and self-accommodation effect of the transformed regions. Previous experimental studies have shown that the presence of pores, particularly at the metal-air interface, can significantly affect the martensitic variant structure, leading to its thinning. This thinning effect has been found to be able to improve the damping performance of the alloy. Experimental observations have indicated that no relief of martensitic variants was found around the metal-air interface, but non-transformed regions were observed. These observations suggest that the metal-air interface in porous materials is not a free surface and plays a crucial role in influencing the martensitic transformation. To further investigate the effect of martensitic variant self-accommodation on different constrained interfaces in porous materials, a three-dimensional phase-field model based on the time dependent Ginzburg-Landau (TDGL) function is proposed in this study. The phase-field model can give a comprehensive understanding of the evolution of martensitic variants and their interaction with the constrained interfaces. Remarkably, the simulation results accord well with the experimental findings, demonstrating the presence of fine martensitic variants near the metal-air interface. The simulations under different interface constraint conditions reveal that increasing the specific surface area of porous materials is an effective strategy to obtain a more refined martensitic variant structure. The system’s total energy is minimized by reducing the strain energy, which leads to the formation of a greater number of fine martensitic variants. This finding suggests that controlling the specific surface area of porous materials can be a promising approach to tailoring the mechanical properties of SMAs for specific applications. In conclusion, the presence of metal-air interface in porous material significantly influences the evolution of the martensitic transformation in SMA. Experimental observations show that the introduction of pore can modify the martensitic variant structure, resulting in improved damping performance. The proposed phase-field model successfully captures the behavior of martensitic variants near constrained interface. The simulation results emphasize the importance of specific surface area in obtaining fine martensitic variant structures. These findings contribute to a more in-depth understanding of the role of porous materials in shaping the properties of SMAs and provide a valuable insight into their design and application in various fields.

Three-dimensional phase field modeling of progressive failure in aramid short fiber reinforced paper

As a kind of synthetic aromatic polyamide, aramid fiber has many outstanding properties, such as high strength, high modulus, etc. The composited paper reinforced by aramid fiber is widely used in honeycombs and its strength is of great concerns. In this paper, a three-dimensional progressive failure phase field model is employed to investigate the failure mechanism of aramid honeycomb paper under microscopic length scale. A mesh mapping technique is used to handle the complex geometry of the material, and an explicit solving scheme is used to improve the solving efficiency. The theory is implemented into ABAQUS through the users’ subroutine VUEL. The interfacial debounding and matrix cracking in the short fiber reinforced composite are captured successfully, and the complicated failure mechanism is explored from the modeling results. The present phase field modeling technique has provided a useful numerical tool for the strength prediction of aramid short fiber reinforced composites.

Three-dimensional phase-field modeling of brittle fracture using an adaptive octree-based scaled boundary finite element approach

In this paper, an adaptive phase-field approach is proposed for three-dimensional fracture modeling in brittle materials. The scaled boundary finite element method (SBFEM) is used to solve the phase-field and elasticity equations in a staggered scheme. This is motivated by the ability of the SBFEM to handle polyhedral elements with hanging nodes straightforwardly. Such elements occur in octree meshes, which facilitate rapid transitions in element size and lend themselves to adaptive refinement. The SBFEM also provides an energy-based error indicator, which follows directly from the semi-analytical solution approach and does not require stress recovery. The error indicator is used to ensure the solution accuracy for both fields and combined with a criterion based on phase-field evolution. The computational cost associated with 3D fracture modeling is further reduced by exploiting the geometric similarity of cells occurring in balanced octree meshes. To validate the proposed method, several classical benchmark problems are solved, resulting in efficient and robust solutions compared to the literature.

Three-dimensional microstructure evolution of Ti–6Al–4V during multi-layer printing: a phase-field simulation

Remelting and grain growth in deposited layers and the columnar-to-equiaxed transition (CET) have a significant impact on microstructural evolution during the multilayer printing process. A three-dimensional phase-field model incorporating a heterogeneous nucleation model was developed to study the microstructural evolution under different scanning strategies and scanning velocities during directed energy deposition (DED) additive manufacturing. An experiment was performed to validate the predicted grain morphologies using the proposed model. The results indicate that undercooling determines the extent of heterogeneous nucleation and controls the CET. In comparison with constitutional undercooling and curvature undercooling, thermal undercooling contributes more to the DED of Ti–6Al–4V. The maximum undercooling occurs on the top layer leading to a higher ratio of equiaxed grain formation. The undercooling increases with the build height, which is beneficial for the CET. The growth direction of the columnar grains is controlled by the size and shape of the melt pool. The bidirectional scanning strategy leads to bent columnar grains as a result of the changing growth direction between adjacent deposited layers. The decrease in the scanning velocity leads to coarsening of the grains.

Local well-posedness of a three-dimensional phase-field model for thrombus and blood flow

In this article we consider a fluid–structure interactions model on a three dimensional bounded domain, that describes the mechanical interaction between blood flow and a thrombus with Hookean elasticity. The interface between the two phases is given by a smooth transition layer, diffuse with a finite thickness. We derive various a priori estimates and prove local well-posedness results using the Faedo–Galerkin method.

Three-dimensional phase-field modeling of temperature-dependent thermal shock-induced fracture in ceramic materials

Current theoretical and experimental methods cannot fully reveal the mechanisms of the rapid and complex thermal shock-induced crack initiation and propagation processes in ceramic materials. Herein, a three-dimensional (3D) coupled thermo-mechanical phase-field model (PFM) is developed for thermal shock-induced fracture with the consideration of the temperature dependence of material properties. Compared with other PFMs, the present model can eliminate the unexpected damage evolution at the initially intact area of materials by introducing a temperature-dependent fracture energy threshold. Both the two-dimensional (2D) and 3D phase-field modeling results of thermal shock-induced fracture show strong agreement with the experimental results. The net-like topologies of thermal shock-induced cracks on the specimen surfaces are captured. Specifically, the crack topologies on the bottom surface (i.e., the first part submerged in water) are significantly different from those on the top surface in 3D cases. These essential findings reveal the mechanism that the tensile part of the strain energy mainly dominates the thermal shock-induced cracking in ceramics.

Simulation of three-dimensional phase field model with LBM method using OpenCL

Simulation of three-dimensional phase field model with LBM method using OpenCL