Phase-field simulations of the effect of temperature and interface for zirconiumδ-hydrides

Hydride precipitation in zirconium cladding materials can damage their integrity and durability.Service temperature and material defects have a significant effect on the dynamic growth of hydrides.In this study,we have developed a phasefield model based on the assumption of elastic behaviour within a specific temperature range(613 K-653 K).This model allows us to study the influence of temperature and interfacial effects on the morphology,stress,and average growth rate of zirconium hydride.The results suggest that changes in temperature and interfacial energy influence the length-to-thickness ratio and average growth rate of the hydride morphology.The ultimate determinant of hydride orientation is the loss of interfacial coherency,primarily induced by interfacial dislocation defects and quantifiable by the mismatch degree q.An escalation in interfacial coherency loss leads to a transition of hydride growth from horizontal to vertical,accompanied by the onset of redirection behaviour.Interestingly,redirection occurs at a critical mismatch level,denoted as qc,and remains unaffected by variations in temperature and interfacial energy.However,this redirection leads to an increase in the maximum stress,which may influence the direction of hydride crack propagation.This research highlights the importance of interfacial coherency and provides valuable insights into the morphology and growth kinetics of hydrides in zirconium alloys.

A Non-Intrusive Stochastic Phase-Field for Fatigue Fracture in Brittle Materials with Uncertainty in Geometry and Material Properties

Understanding the probabilistic nature of brittle materials due to inherent dispersions in their mechanical properties is important to assess their reliability and safety for sensitive engineering applications.This is all the more important when elements composed of brittle materials are exposed to dynamic environments,resulting in catastrophic fatigue failures.The authors propose the application of a non-intrusive polynomial chaos expansion method for probabilistic studies on brittle materials undergoing fatigue fracture when geometrical parameters and material properties are random independent variables.Understanding the probabilistic nature of fatigue fracture in brittle materials is crucial for ensuring the reliability and safety of engineering structures subjected to cyclic loading.Crack growth is modelled using a phase-field approach within a finite element framework.For modelling fatigue,fracture resistance is progressively degraded by modifying the regularised free energy functional using a fatigue degradation function.Number of cycles to failure is treated as the dependent variable of interest and is estimated within acceptable limits due to the randomness in independent properties.Multiple 2D benchmark problems are solved to demonstrate the ability of this approach to predict the dependent variable responses with significantly fewer simulations than the Monte Carlo method.This proposed approach can accurately predict results typically obtained through 105 or more runs in Monte Carlo simulations with a reduction of up to three orders of magnitude in required runs.The independent random variables’sensitivity to the system response is determined using Sobol’indices.The proposed approach has low computational overhead and can be useful for computationally intensive problems requiring rapid decision-making in sensitive applications like aerospace,nuclear and biomedical engineering.The technique does not require reformulating existing finite element code and can perform the stochastic study by direct pre/post-processing.

Dendritic tip selection during solidification of alloys:Insights from phase-field simulations

The effect of undercooling DT and the interface energy anisotropy parameter e4 on the shape of the equiaxed dendritic tip has been investigated by using a quantitative phase-field model for solidification of binary alloys.It was found that the tip radius r increases and the tip shape amplitude coefficient A4 decreases with the increase of the fitting range for all cases.The dendrite tip shape selection parameter sdecreases and then stabilizes with the increase of the fitting range,and sincreases with the increase of e4.The relationship between sand e4 follows a power-law function sµea 4,and a is independent of DT but dependent on the fitting range.Numerical results demonstrate that the predicted sis consistent with the curve of microscopic solvability theory(MST)for e4<0.02,and sobtained from our phase-field simulations is sensitive to the undercooling when e4 is fixed.

A phase-field model for simulating the propagation behavior of mixed-mode cracks during the hydraulic fracturing process in fractured reservoirs

A novel phase-field model for the propagation of mixed-mode hydraulic fractures,characterized by the formation of mixed-mode fractures due to the interactions between fluids and solids,is proposed.In this model,the driving force for the phase field consists of both tensile and shear components,with the fluid contribution primarily manifesting in the tension driving force.The displacement and pressure are solved simultaneously by an implicit method.The numerical solution's iterative format is established by the finite element discretization and Newton-Raphson(NR)iterative methods.The correctness of the model is verified through the uniaxial compression physical experiments on fluid-pressurized rocks,and the limitations of the hydraulic fracture expansion phase-field model,which only considers mode I fractures,are revealed.In addition,the influence of matrix mode II fracture toughness value,natural fracture mode II toughness value,and fracturing fluid injection rate on the hydraulic fracture propagation in porous media with natural fractures is studied.

Phase-Field Simulation of Sintering Process:A Review

Sintering,a well-established technique in powder metallurgy,plays a critical role in the processing of high melting point materials.A comprehensive understanding of structural changes during the sintering process is essential for effective product assessment.The phase-field method stands out for its unique ability to simulate these structural transformations.Despite its widespread application,there is a notable absence of literature reviews focused on its usage in sintering simulations.Therefore,this paper addresses this gap by reviewing the latest advancements in phase-field sintering models,covering approaches based on energy,grand potential,and entropy increase.The characteristics of various models are extensively discussed,with a specific emphasis on energy-based models incorporating considerations such as interface energy anisotropy,tensor-form diffusion mechanisms,and various forms of rigid particle motion during sintering.Furthermore,the paper offers a concise summary of phase-field sintering models that integrate with other physical fields,including stress/strain fields,viscous flow,temperature field,and external electric fields.In conclusion,the paper provides a succinct overview of the entire content and delineates potential avenues for future research.

Fourth-order phase-field modeling for brittle fracture in piezoelectric materials

Failure analyses of piezoelectric structures and devices are of engineering and scientific significance.In this paper,a fourth-order phase-field fracture model for piezoelectric solids is developed based on the Hamilton principle.Three typical electric boundary conditions are involved in the present model to characterize the fracture behaviors in various physical situations.A staggered algorithm is used to simulate the crack propagation.The polynomial splines over hierarchical T-meshes(PHT-splines)are adopted as the basis function,which owns the C1continuity.Systematic numerical simulations are performed to study the influence of the electric boundary conditions and the applied electric field on the fracture behaviors of piezoelectric materials.The electric boundary conditions may influence crack paths and fracture loads significantly.The present research may be helpful for the reliability evaluation of the piezoelectric structure in the future applications.

Recent research progress on the phase-field model of microstructural evolution during metal solidification

Solidification structure is a key aspect for understanding the mechanical performance of metal alloys,wherein composition and casting parameters considerably influence solidification and determine the unique microstructure of the alloys.By following the principle of free energy minimization,the phase-field method eliminates the need for tracking the solid/liquid phase interface and has greatly accelerated the research and development efforts geared toward optimizing metal solidification microstructures.The recent progress in the application of phasefield simulation to investigate the effect of alloy composition and casting process parameters on the solidification structure of metals is summarized in this review.The effects of several typical elements and process parameters,including carbon,boron,silicon,cooling rate,pulling speed,scanning speed,anisotropy,and gravity,on the solidification structure are discussed.The present work also addresses the future prospects of phase-field simulation and aims to facilitate the widespread applications of phase-field approaches in the simulation of microstructures during solidification.

Phase-field simulation of lack-of-fusion defect and grain growth during laser powder bed fusion of Inconel 718

The anisotropy of the structure and properties caused by the strong epitaxial growth of grains during laser powder bed fusion(L-PBF)significantly affects the mechanical performance of Inconel 718 alloy components such as turbine disks.The defects(lack-of-fusion Lo F)in components processed via L-PBF are detrimental to the strength of the alloy.The purpose of this study is to investigate the effect of laser scanning parameters on the epitaxial grain growth and LoF formation in order to obtain the parameter space in which the microstructure is refined and LoF defect is suppressed.The temperature field of the molten pool and the epitaxial grain growth are simulated using a multiscale model combining the finite element method with the phase-field method.The LoF model is proposed to predict the formation of LoF defects resulting from insufficient melting during L-PBF.Defect mitigation and grain-structure control during L-PBF can be realized simultaneously in the model.The simulation shows the input laser energy density for the as-deposited structure with fine grains and without LoF defects varied from 55.0–62.5 J·mm^(-3)when the interlayer rotation angle was 0°–90°.The optimized process parameters(laser power of 280 W,scanning speed of 1160 mm·s^(-1),and rotation angle of 67°)were computationally screened.In these conditions,the average grain size was 7.0μm,and the ultimate tensile strength and yield strength at room temperature were(1111±3)MPa and(820±7)MPa,respectively,which is 8.8%and10.5%higher than those of reported.The results indicating the proposed multiscale computational approach for predicting grain growth and Lo F defects could allow simultaneous grain-structure control and defect mitigation during L-PBF.

Improved Staggered Algorithm for Phase-Field Brittle Fracture with the Local Arc-Length Method

The local arc-length method is employed to control the incremental loading procedure for phase-field brittle fracture modeling.An improved staggered algorithm with energy and damage iterative tolerance convergence criteria is developed based on the residuals of displacement and phase-field.The improved staggered solution scheme is implemented in the commercial software ABAQUS with user-defined element subroutines.The layered system of finite elements is utilized to solve the coupled elastic displacement and phase-field fracture problem.A one-element benchmark test compared with the analytical solution was conducted to validate the feasibility and accuracy of the developed method.Our study shows that the result calculated with the developed method does not depend on the selected size of loading increments.The results of several numerical experiments show that the improved staggered algorithm is efficient for solving the more complex brittle fracture problems.

Parameter calculation and result storage for phase-field simulation in α-Mg dendrite growth of Mg-5-wt% Zn alloy

Parameter calculation and result storage, as two necessary steps in phase-field simulation play an important role in ensuring the accuracy of simulation results. A strategy of parameter calculation and result storage is presented for phase-field simulation in α-Mg dendrite growth of Mg-5-wt% Zn alloy under isothermal solidification. Based on the phase diagram and empirical formulas, key parameters of the phase-field model, such as equilibrium partition coefficient k, liquidus slope m, solutal diffusion coefficient in liquid Dl, and solutal diffusion coefficient in solid Ds, can be obtained.Both structured grid method and structured point method can be used to store simulation results, but using the latter method will reduce about 60% storage space and 37.5% storage time compared with the former. Finally, convergent simulation results of α-Mg dendrite growth are obtained and they are in good agreement with the experimental results about optical micrograph, which verify the accuracy of parameters and stability of storage method.

Phase-Field Simulation of δ Hydride Precipitation with Interfacial Anisotropy

Previous studies ofδhydride in zirconium alloys have mainly assumed an isotropic interface.In practice,the difference in crystal structure at the interface between the matrix phase and the precipitate phase results in an anisotropic interface.With the purpose of probing the real evolution ofδhydrides,this paper couples an anisotropy function in the interfacial energy and interfacial mobility.The influence of anisotropic interfacial energy and interfacial mobility on the morphology ofδhydride precipitation was investigated using the phase-field method.The results show that the isotropy hydride precipitates a slate-like morphology,and the anisotropicδhydride precipitates at the semi-coherent and non-coherent interfaces exhibited parallelogram-like and needle-like,which is consistent with the actual experimental morphology.Compared with the coherent interface,the semi-coherent or non-coherent interface adjusts the lattice mismatch,resulting in lower gradient energy that is more consistent with the true interfacial state.Simultaneously,an important chain of relationships is proposed,in the range of I_(x)<I_(y)<1.5I_(x)(I_(y)<I_(x) or I_(y)>1.5I_(x)),with the increase of the anisotropic mobility I_(y) in the y-axis,the gradient energy increases(decreases),the tendency of the non-coherent(semi-coherent)relationship at the interface,and the precipitation rate of hydride decreases(increases).Furthermore,the inhomogeneous stress distribution around the hydride leads to a localized enrichment of the hydrogen concentration,producing a hydride tip.The study of interfacial anisotropy is informative for future studies ofδhydride precipitation orientation and properties.

Three dimensional phase-field simulation for non-isothermal binary alloy solidification: Comparison with LKT theory

Using the advanced algorithm combining parallel computing,adaptive mesh re-griding and multigrid methods,quantitative 3D phase-field simulations of non-isothermal solidification of binary alloy were carried out.The 3D phase-field simulation results were compared with the analytical LKT(Lipton,Kurz and Trivedi)theory.For comparison,the simulation and analytical results for 2D cases were also given.The 3D phase-field simulation results support the transport portion of the LKT theory.However,the tip radius and tip velocity predicted by the simulations are not in good agreement with the LKT theory over the whole range of undercooling.The stability parameter calculated from phase-field simulations varies significantly with the Peclet number,indicating that the stability criterion,which assumes that the stability parameter is constant,is invalid.

A new mixed-mode phase-field model for crack propagation of brittle rock

Study on crack propagation process of brittle rock is of most significance for cracking-arrest design and cracking-network optimization in rock engineering.Phase-field model(PFM)has advantages of simplicity and high convergence over the common numerical methods(e.g.finite element method,discrete element method,and particle manifold method)in dealing with three-dimensional and multicrack problems.However,current PFMs are mainly used to simulate mode-I(tensile)crack propagation but difficult to effectively simulate mode-II(shear)crack propagation.In this paper,a new mixed-mode PFM is established to simulate both mode-I and mode-II crack propagation of brittle rock by distinguishing the volumetric elastic strain energy and deviatoric elastic strain energy in the total elastic strain energy and considering the effect of compressive stress on mode-II crack propagation.Numerical solution method of the new mixed-mode PFM is proposed based on the staggered solution method with self-programmed subroutines UMAT and HETVAL of ABAQUS software.Three examples calculated using different PFMs as well as test results are presented for comparison.The results show that compared with the conventional PFM(which only simulates the tensile wing crack but not mode-II crack propagation)and the modified mixed-mode PFM(which has difficulty in simulating the shear anti-wing crack),the new mixed-mode PFM can successfully simulate the whole trajectories of mixed-mode crack propagation(including the tensile wing crack,shear secondary crack,and shear anti-wing crack)and mode-II crack propagation,which are close to the test results.It can be further extended to simulate multicrack propagation of anisotropic rock under multi-field coupling loads.

Phase-field simulations of forced flow effect on dendritic growth perpendicular to flow

The effect of supercooled melt forced laminar flow at low Reynolds Number on dendritic growth perpendicular to melt flow direction was investigated with the phase-field method by incorporating melt convection and thermal noise under non-isothermal condition. By taking the dendritic growth of high pure succinonitrile （SCN） supercooled melt as an example, side-branching shape difference of melts with flow and without flow was analyzed. Relationships among supercooled melt inflow velocity, deflexion angle of dendritic arm and dendritic tip growth velocity were studied. Results show that the melt inflow velocity has few effects on the dendritic tip growth velocity. A formula of relationship between the velocity of the melt in front of primary dendritic tip and the dendritic growth time was deduced, and the calculated result was in quantitative agreement with the simulation result.

Microscopic phase-field study on directional coarsening mechanism caused by interaction between precipitates in Ni-Al-V alloy

A microscopic phase-field model was used to investigate a directional coarsening mechanism caused by the anisotropic growth of long period stacking and different effects of phases on precipitation in Ni-Al-V alloy.The results show that DO22 mainly coarsens along its short axis,which may press the neighboring L12,leading to the interaction among atoms.Diffusion channels of Al are formed in the direction where the mismatch between γ＇ and γ reduces;the occupation probabilities are anisotropic in space;and direction coarsening of L12 occurs finally.With a rise of ageing temperature,phases appear later and DO22 is much later at a higher temperature,the average occupation probabilities of Al and V reduce,and Al changes more than V.

Phase-field study on competition precipitation process of Ni-Al-V alloy

With microscopic phase-field kinetic model, atomic-scale computer simulation program for the precipitation sequence and microstructure evolution of the ordered intermetallic compound γ＇ and θ in ternary Ni75AlxV25-x alloy were studied. The simulation results show that Al concentration has important effects on the precipitation sequence. When Al concentration in Ni75AlxV25-x alloy is low, 0（Ni3V） ordered phase will be firstly precipitated, followed by γ＇（Ni3Al） ordered phase. With Al concentration increasing, θ and γ＇ ordered phases are simultaneously precipitated. With A1 concentration further increasing, γ＇ ordered phase is firstly precipitated, followed by θ ordered phase. There is a competition relationship between θ and γ＇ ordered phases during growth and coarsening process. No matter which first precipitates, θ ordered phase always occupies advantage in the competition process of coarsening, thus, the microstructure with preferred orientation is formed.

Comparative analysis of isothermal and non-isothermal solidification of binary alloys using phase-field model

Based on the entropy function, a two-dimensional phase field model of binary alloys was established. Meanwhile, an explicit difference method with uniform grid was adopted to solve the phase field and solute field controlled equations. And the alternating direction implicit(ADI) algorithm for solving temperature field controlled equation was also employed to avoid the restriction of time step. Some characteristics of the Ni-Cu alloy were captured in the process of non-isothermal solidification, and the comparative analysis of the isothermal and the non-isothermal solidification was investigated. The simulation results indicate that the non-isothermal model is favorable to simulate the real solidification process of binary alloys, and when the thermal diffusivity decreases, the non-isothermal phase-field model is gradually consistent with the isothermal phase-field model.

Phase-field modeling of dendritic growth under forced flow based on adaptive finite element method

A mathematical model combined projection algorithm with phase-field method was applied. The adaptive finite element method was adopted to solve the model based on the non-uniform grid, and the behavior of dendritic growth was simulated from undercooled nickel melt under the forced flow. The simulation results show that the asymmetry behavior of the dendritic growth is caused by the forced flow. When the flow velocity is less than the critical value, the asymmetry of dendrite is little influenced by the forced flow. Once the flow velocity reaches or exceeds the critical value, the controlling factor of dendrite growth gradually changes from thermal diffusion to convection. With the increase of the flow velocity, the deflection angle towards upstream direction of the primary dendrite stem becomes larger. The effect of the dendrite growth on the flow field of the melt is apparent. With the increase of the dendrite size, the vortex is present in the downstream regions, and the vortex region is gradually enlarged. Dendrite tips appear to remelt. In addition, the adaptive finite element method can reduce CPU running time by one order of magnitude compared with uniform grid method, and the speed-up ratio is proportional to the size of computational domain.

Phase-field simulation of forced flow effect on random preferred growth direction of multiple grains

The random distribution problem of dendrite preferred growth direction was settled by random grid method.This method was used to study the influence of forced laminar flow effect on multiple grains during solidification.Taking high pure succinonitrile （SCN） undercooled melt as an example,the forced laminar flow effect on multiple grains was studied by phase-field model of single grain which coupled with flow equations at non-isothermal condition.The simulation results show that the random grid method can reasonably settle the problem of random distribution and is more effective.When the solid fraction is relatively low,melt particles flow around the downstream side of dendrite,and the flow velocity between two dendrite arms becomes high.At the stage of solidification time less than 1800Δt,every dendrite grows freely;the upstream dendrites are stronger than the downstream ones.The higher the melt flow rate,the higher the solid fraction.However,when the solid fraction is relatively high,the dendrite arm intertwins and only a little residual melt which is not encapsulated can flow;the solid fraction will gradually tend to equal to solid fraction of melt without flow.

Simulation of pre-precipitation in Ni_(75)Al_(14)Mo_(11) alloy by microscopic phase-field model

The early precipitation process of Ni（75）Al（14）Mo（11） alloy was simulated by microscopic phase-field model at different temperatures.The microstructure of the alloy,the precipitation time of Llo structure and occupation probability of the three kinds of atoms were investigated.It is indicated that the non-stoichiometric Ll0（Ⅰ/Ⅱ） phases are found in the precipitation process.With the temperature increasing,the appearance time of Ll0 is brought forward.The Ll0（Ⅱ） structure always precipitates earlier than the Ll0（Ⅰ） structure.Compared with lower temperature,higher temperature brings the formation time of Ll0 phase forward and makes Ll0 phase have a higher order degree.But lower temperature shortens the process time of the Ll0 phase to the Ll2 phase.Al and Mo atoms tend to occupy γ site,Ni atom tends to occupy a and β sites.At the same temperature,Al atom has stronger occupation ability than Mo atom in the same site.Ni,Al and Mo collectively form the composited Ll2 structure.