Phase field method offers the prospect of being able to perform realistic numerical experiments on dendrite growth in metallic systems. In this study, the growth process of multiple dendrites in AI-2-mole-%-Si binary ...Phase field method offers the prospect of being able to perform realistic numerical experiments on dendrite growth in metallic systems. In this study, the growth process of multiple dendrites in AI-2-mole-%-Si binary alloy under isothermal solidification was simulated using phase field model. The simulation results showed the impingement of arbitrarily oriented crystals and the competitive growth among the grains during solidification. With the increase of growing time, the grains begin to coalesce and impinge the adjacent grains. When the dendrites start to impinge, the dendrite growth is obviously inhibited.展开更多
Liquid permeability of the mushy zone is important for porosity formation during the solidification process. In order to investigate the permeability of the mushy zone, an integrated model was developed by combining t...Liquid permeability of the mushy zone is important for porosity formation during the solidification process. In order to investigate the permeability of the mushy zone, an integrated model was developed by combining the phase field model and computational fluid dynamics (CFD) model. The three-dimensional multigrain dendrite morphology was obtained by using the phase field model. Subsequently, the computer-aided design (CAD) geometry and mesh were generated based on calculated dendrite morphologies. Finally, the permeability of the dendritic mushy zone was obtained by solving the Navier-Stokes and continuity equations in ANSYS Fluent software. As an example, the dendritic mushy zone permeability of Al-4.5wt%Cu alloy and its relationship with the solid fractions were studied in detail. The predicted permeability data can be input to the solidification model on a greater length scale for macro segregation and porosity simulations.展开更多
The quantitative phase-field simulations were reviewed on the processes of solidification of pure metals and alloys.The quantitative phase-field equations were treated in a diffuse thin-interface limit,which enabled t...The quantitative phase-field simulations were reviewed on the processes of solidification of pure metals and alloys.The quantitative phase-field equations were treated in a diffuse thin-interface limit,which enabled the quantitative links between interface dynamics and model parameters in the quasi-equilibrium simulations.As a result,the quantitative modeling is more effective in dealing with microstructural pattern formation in the large scale simulations without any spurious kinetic effects.The development of the quantitative phase-field models in modeling the formation of microstructures such as dendritic structures,eutectic lamellas,seaweed morphologies,and grain boundaries in different solidified conditions was also reviewed with the purpose of guiding to find the new prospect of applications in the quantitative phase-field simulations.展开更多
In this study, the phase field method was used to study the multi-controlling factors of dendrite growth in directional solidification. The effects of temperature gradient, propelling velocity, thermal disturbance and...In this study, the phase field method was used to study the multi-controlling factors of dendrite growth in directional solidification. The effects of temperature gradient, propelling velocity, thermal disturbance and growth orientation angle on the growth morphology of the dendritic growth in the solid/liquid interface were discussed. It is found that the redistribution of solute leads to multilevel cavity and multilevel fusion to form multistage solute segregation, and the increase of temperature gradient and propelling velocity can accelerate the dendrite growth of directional solidification, and also make the second dendrites more developed, which reduces the primary distance and the solute segregation. When the temperature gradient is large, the solid-liquid interface will move forward in a flat interface mode,and the thermal disturbance does not affect the steady state behavior of the directionally solidified dendrite tip. It only promotes the generation and growth of the second dendrites and forms the asymmetric dendrite. Meanwhile, it is found that the inclined dendrite is at a disadvantage in the competitive growth compared to the normal dendrite, and generally it will disappear. When the inclination angle is large, the initial primary dendrite may be eliminated by its secondary or third dendrite.展开更多
Single dendrite and multi-dendrite growth for Al-2 mol pct Si alloy during isothermal solidification are simulated by phase field method. In the case of single equiaxed dendrite growth, the secondary and the necking p...Single dendrite and multi-dendrite growth for Al-2 mol pct Si alloy during isothermal solidification are simulated by phase field method. In the case of single equiaxed dendrite growth, the secondary and the necking phenomenon can be observed. For multi-dendrite growth, there exists the competitive growth among the dendrites during solidification. As solidification proceeds, growing and coarsening of the primary arms occurs, together with the branching and coarsening of the secondary arms. When the diffusion fields of dendrite tips come into contact with those of the branches growing from the neighboring dendrites, the dendrites stop growing and being to ripen and thicken.展开更多
All the quantitative phase field models try to get rid of the artificial factors of solutal drag, interface diffusion and interface stretch in the diffuse interface. These artificial non-equilibrium effects due to the...All the quantitative phase field models try to get rid of the artificial factors of solutal drag, interface diffusion and interface stretch in the diffuse interface. These artificial non-equilibrium effects due to the introducing of diffuse interface are analysed based on the thermodynamic status across the diffuse interface in the quantitative phase field model of binary alloys. Results indicate that the non-equilibrium effects are related to the negative driving force in the local region of solid side across the diffuse interface. The negative driving force results from the fact that the phase field model is derived from equilibrium condition but used to simulate the non-equilibrium solidification process. The interface thickness dependence of the non-equilibrium effects and its restriction on the large scale simulation are also discussed.展开更多
The ring-banded spherulite is a special morphology of polymer crystals and has attracted considerable attention over recent decades. In this study, a new phase field model with polymer characteristics is established t...The ring-banded spherulite is a special morphology of polymer crystals and has attracted considerable attention over recent decades. In this study, a new phase field model with polymer characteristics is established to investigate the emergence and formation mechanism of the ring-banded spherulites of crystalline polymers. The model consists of a nonconserved phase field representing the phase transition and a temperature field describing the diffusion of the released latent heat. The corresponding model parameters can be obtained from experimentally accessible material parameters.Two-dimensional calculations are carried out for the ring-banded spherulitic growth of polyethylene film under a series of crystallization temperatures. The results of these calculations demonstrate that the formation of ring-banded spherulites can be triggered by the self-generated thermal field. Moreover, some temperature-dependent characteristics of the ring-banded spherulites observed in experiments are reproduced by simulations, which may help to study the effects of crystallization temperature on the ring-banded structures.展开更多
For solid-fluid interaction, one of the phase-density equations in diffuse interface models is degenerated to a "0 = 0" equation when the volume fraction of a certain phase takes the value of zero or unity. ...For solid-fluid interaction, one of the phase-density equations in diffuse interface models is degenerated to a "0 = 0" equation when the volume fraction of a certain phase takes the value of zero or unity. This is because the conservative variables in phasedensity equations include volume fractions. The degeneracy can be avoided by adding an artificial quantity of another material into the pure phase. However, nonphysical waves,such as shear waves in fluids, are introduced by the artificial treatment. In this paper,a transport diffuse interface model, which is able to treat zero/unity volume fractions, is presented for solid-fluid interaction. In the proposed model, a new formulation for phase densities is derived, which is unrelated to volume fractions. Consequently, the new model is able to handle zero/unity volume fractions, and nonphysical waves caused by artificial volume fractions are prevented. One-dimensional and two-dimensional numerical tests demonstrate that more accurate results can be obtained by the proposed model.展开更多
A novel approach based on the quantitative phase field model was proposed to calculate the interface mobility and applied to the α/β interface of a ternary Ti-6Al-4V alloy.Phase field simulations indicate that the h...A novel approach based on the quantitative phase field model was proposed to calculate the interface mobility and applied to the α/β interface of a ternary Ti-6Al-4V alloy.Phase field simulations indicate that the higher interface mobility leads to the faster transformation rate,but only a unique value of interface mobility matches the diffusion equation under the diffusion-controlled condition.By comparing the transformation kinetics from phase field simulations with that from classical diffusion equation,the interface mobility at different temperatures can be obtained.The results show that the calculated interface mobility increases with increasing temperature and accords with Arrhenius equation very well.展开更多
A new phase field method for two-dimensional simulations of binary alloy solidification was studied. A model basing on solute conservative in every unit was developed for solving the solute diffusion equation during s...A new phase field method for two-dimensional simulations of binary alloy solidification was studied. A model basing on solute conservative in every unit was developed for solving the solute diffusion equation during solidification. Two-dimensional computations were performed for ideal solutions and Ni-Cu dendritic growth into an isothermal and highly supersaturated liquid phase.展开更多
In this paper,we study linearly first and second order in time,uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the ...In this paper,we study linearly first and second order in time,uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel"scalar auxiliary variable"(SAV)approach,a new developed efficient and accurate method for a large class of gradient flows.The schemes are based on the first order Euler method and the second order backward differential formulas(BDF2)for time discretization,and finite element methods for space discretization.The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps.Various numerical experiments are presented to validate the stability and accuracy of the proposed schemes.展开更多
We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification.Here we focus on Stefan problems,and their quasi-static variants,with ...We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification.Here we focus on Stefan problems,and their quasi-static variants,with applications to crystal growth.New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.展开更多
The temporal interface microstructures and diffusions in the diffusion couples with the mutual interactions of the temperature gradient, concentration difference and initial aging time of the alloys are studied by pha...The temporal interface microstructures and diffusions in the diffusion couples with the mutual interactions of the temperature gradient, concentration difference and initial aging time of the alloys are studied by phase-field simulation, and the diffusion couples are produced by the initial aged spinodal alloys with different compositions. Temporal composition evolution and volume fraction of the separated phase indicate the element diffusion direction through the interface under the temperature gradient. The increased temperature gradient induces a wide single-phase region on two sides of the interface.The uphill diffusion proceeds through the interface, no matter whether the diffusion direction is up or down with respect to the temperature gradient. For an alloy with short initial aging time, phase transformation accompanying the interdiffusion results in the straight interface with the single-phase regions on both sides. Compared with the temperature gradient,composition difference of diffusion couple and initial aging time of the alloy show greater effects on diffusion and interface microstructure.展开更多
This paper applies a phase field model for polycrystalline solidification in binary alloys to simulate the formation and growth of the columnar dendritic array under the isothermal and constant cooling conditions. The...This paper applies a phase field model for polycrystalline solidification in binary alloys to simulate the formation and growth of the columnar dendritic array under the isothermal and constant cooling conditions. The solidification process and microsegregation in the mushy zone are analysed in detail. It is shown that under the isothermal condition solidification will stop after the formation of the mushy zone, but dendritic coarsening will progress continuously, which results in the decrease of the total interface area. Under the constant cooling condition the mushy zone will solidify and coarsen simultaneously. For the constant cooling solidification, microsegregation predicted by a modified Brody- Flemings model is compared with the simulation results. It is found that the Fourier number which characterizes microsegregation is different for regions with different microstructures. Dendritic coarsening and the larger area of interface should account for the enhanced Fourier number in the region with well developed second dendritic arms.展开更多
The phase-field method has emerged as the method of choice for the description of microstructure evolution and phase transitions in metallic materials.Following general thermodynamic laws a set of evolution equations ...The phase-field method has emerged as the method of choice for the description of microstructure evolution and phase transitions in metallic materials.Following general thermodynamic laws a set of evolution equations for the structural variables of the system,the so called phase-fields,are derived.The paper reviews shortly the theoretical background of the multi-phase-field.Different examples demonstrating the applicability of the method to technical steels will be presented ranging from deformation of the dendritic strand shell during peritectic transformation,grain growth in Austenite to stress driven growth of Pearlite.展开更多
The effects of rotation and gravity on an electro-magneto-thermoelastic medium with diffusion and voids in a generalized thermoplastic half-space are studied by using the Lord-Shulman (L-S) model and the dual-phase-la...The effects of rotation and gravity on an electro-magneto-thermoelastic medium with diffusion and voids in a generalized thermoplastic half-space are studied by using the Lord-Shulman (L-S) model and the dual-phase-lag (DPL) model. The analytical solutions for the displacements, stresses, temperature, diffusion concentration, and volume fraction field with different values of the magnetic field, the rotation, the gravity, and the initial stress are obtained and portrayed graphically. The results indicate that the effects of gravity, rotation, voids, diffusion, initial stress, and electromagnetic field are very pronounced on the physical properties of the material.展开更多
This paper reviews differences between the deterministic(sharp and diffuse)and statistical models of the interphase region between the two-phases.In the literature this region is usually referred to as the(macroscopic...This paper reviews differences between the deterministic(sharp and diffuse)and statistical models of the interphase region between the two-phases.In the literature this region is usually referred to as the(macroscopic)interface.Therein,the mesoscopic interface that is defined at the molecular level and agitated by the thermal fluctuations is found with nonzero probability.For this reason,in this work,the interphase region is called the mesoscopic intermittency/transition region.To this purpose,the first part of the present work gives the rationale for introduction of the mesoscopic intermittency region statistical model.It is argued that classical(deterministic)sharp and diffuse models do not explain the experimental and numerical results presented in the literature.Afterwards,it is elucidated that a statistical model of the mesoscopic intermittency region(SMIR)combines existing sharp and diffuse models into a single coherent framework and explains published experimental and numerical results.In the second part of the present paper,the SMIR is used for the first time to predict equilibrium and nonequilibrium two-phase flow in the numerical simulation.To this goal,a two-dimensional rising gas bubble is studied;obtained numerical results are used as a basis to discuss differences between the deterministic and statistical models showing the statistical description has a potential to account for the physical phenomena not previously considered in the computer simulations.展开更多
We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gra...We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gradient recovery techniques,we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters.Numerical examples and applications to the topological analysis of point clouds are also presented.展开更多
基金supported by the Doctor Foundational Research Project in Shenyang Ligong University(Serial Number:0010).
文摘Phase field method offers the prospect of being able to perform realistic numerical experiments on dendrite growth in metallic systems. In this study, the growth process of multiple dendrites in AI-2-mole-%-Si binary alloy under isothermal solidification was simulated using phase field model. The simulation results showed the impingement of arbitrarily oriented crystals and the competitive growth among the grains during solidification. With the increase of growing time, the grains begin to coalesce and impinge the adjacent grains. When the dendrites start to impinge, the dendrite growth is obviously inhibited.
基金financially supported by the National Key Research and Development Program of China(No.2016YFB0700503)National Natural Science Foundation of China(No.51701013)Beijing Laboratory of Metallic Materials and Processing for Modern Transportation
文摘Liquid permeability of the mushy zone is important for porosity formation during the solidification process. In order to investigate the permeability of the mushy zone, an integrated model was developed by combining the phase field model and computational fluid dynamics (CFD) model. The three-dimensional multigrain dendrite morphology was obtained by using the phase field model. Subsequently, the computer-aided design (CAD) geometry and mesh were generated based on calculated dendrite morphologies. Finally, the permeability of the dendritic mushy zone was obtained by solving the Navier-Stokes and continuity equations in ANSYS Fluent software. As an example, the dendritic mushy zone permeability of Al-4.5wt%Cu alloy and its relationship with the solid fractions were studied in detail. The predicted permeability data can be input to the solidification model on a greater length scale for macro segregation and porosity simulations.
基金supported by National Natural Science Foundation of China(No.51174177)the Fund of the State Key Solidification Laboratory of Solidification Processing in Northwestern Polytechnical University(No.SKLSP 201714)
文摘The quantitative phase-field simulations were reviewed on the processes of solidification of pure metals and alloys.The quantitative phase-field equations were treated in a diffuse thin-interface limit,which enabled the quantitative links between interface dynamics and model parameters in the quasi-equilibrium simulations.As a result,the quantitative modeling is more effective in dealing with microstructural pattern formation in the large scale simulations without any spurious kinetic effects.The development of the quantitative phase-field models in modeling the formation of microstructures such as dendritic structures,eutectic lamellas,seaweed morphologies,and grain boundaries in different solidified conditions was also reviewed with the purpose of guiding to find the new prospect of applications in the quantitative phase-field simulations.
基金financially supported by the National Natural Science Foundation of China(NSFC)under grant Nos.51774254,51774253,U1610123,51574207,51574206the Science and Technology Major Project of Shanxi Province under grant No.MC2016-06
文摘In this study, the phase field method was used to study the multi-controlling factors of dendrite growth in directional solidification. The effects of temperature gradient, propelling velocity, thermal disturbance and growth orientation angle on the growth morphology of the dendritic growth in the solid/liquid interface were discussed. It is found that the redistribution of solute leads to multilevel cavity and multilevel fusion to form multistage solute segregation, and the increase of temperature gradient and propelling velocity can accelerate the dendrite growth of directional solidification, and also make the second dendrites more developed, which reduces the primary distance and the solute segregation. When the temperature gradient is large, the solid-liquid interface will move forward in a flat interface mode,and the thermal disturbance does not affect the steady state behavior of the directionally solidified dendrite tip. It only promotes the generation and growth of the second dendrites and forms the asymmetric dendrite. Meanwhile, it is found that the inclined dendrite is at a disadvantage in the competitive growth compared to the normal dendrite, and generally it will disappear. When the inclination angle is large, the initial primary dendrite may be eliminated by its secondary or third dendrite.
基金financially supported by the Educational Department of Liaoning Province (No.20060744)the Shenyang Nurturing Young Scientific Technological Talents Items (No.1081230-1-00)
文摘Single dendrite and multi-dendrite growth for Al-2 mol pct Si alloy during isothermal solidification are simulated by phase field method. In the case of single equiaxed dendrite growth, the secondary and the necking phenomenon can be observed. For multi-dendrite growth, there exists the competitive growth among the dendrites during solidification. As solidification proceeds, growing and coarsening of the primary arms occurs, together with the branching and coarsening of the secondary arms. When the diffusion fields of dendrite tips come into contact with those of the branches growing from the neighboring dendrites, the dendrites stop growing and being to ripen and thicken.
基金supported by the fund of the State Key Laboratory of Solidification Processing in NWPU,China (Grants Nos. 17-TZ-2007,03-TP-2008,and 24-TZ-2009)the Doctorate Foundation of Northwestern Polytechnical University
文摘All the quantitative phase field models try to get rid of the artificial factors of solutal drag, interface diffusion and interface stretch in the diffuse interface. These artificial non-equilibrium effects due to the introducing of diffuse interface are analysed based on the thermodynamic status across the diffuse interface in the quantitative phase field model of binary alloys. Results indicate that the non-equilibrium effects are related to the negative driving force in the local region of solid side across the diffuse interface. The negative driving force results from the fact that the phase field model is derived from equilibrium condition but used to simulate the non-equilibrium solidification process. The interface thickness dependence of the non-equilibrium effects and its restriction on the large scale simulation are also discussed.
基金Project supported by the National Key Basic Research Program of China(Grant No.2012CB025903)the National Natural Science Foundation of China(Grant No.11402210)+1 种基金the Northwestern Polytechnical University Foundation for Fundamental Research(Grant No.JCY20130141)the Ministry of Education Fund for Doctoral Students Newcomer Awards of China
文摘The ring-banded spherulite is a special morphology of polymer crystals and has attracted considerable attention over recent decades. In this study, a new phase field model with polymer characteristics is established to investigate the emergence and formation mechanism of the ring-banded spherulites of crystalline polymers. The model consists of a nonconserved phase field representing the phase transition and a temperature field describing the diffusion of the released latent heat. The corresponding model parameters can be obtained from experimentally accessible material parameters.Two-dimensional calculations are carried out for the ring-banded spherulitic growth of polyethylene film under a series of crystallization temperatures. The results of these calculations demonstrate that the formation of ring-banded spherulites can be triggered by the self-generated thermal field. Moreover, some temperature-dependent characteristics of the ring-banded spherulites observed in experiments are reproduced by simulations, which may help to study the effects of crystallization temperature on the ring-banded structures.
基金Project supported by the National Natural Science Foundation of China(Nos.11702029,11771054,U1730118,91852207,and 11801036)the China Postdoctoral Science Foundation(No.2016M600967)
文摘For solid-fluid interaction, one of the phase-density equations in diffuse interface models is degenerated to a "0 = 0" equation when the volume fraction of a certain phase takes the value of zero or unity. This is because the conservative variables in phasedensity equations include volume fractions. The degeneracy can be avoided by adding an artificial quantity of another material into the pure phase. However, nonphysical waves,such as shear waves in fluids, are introduced by the artificial treatment. In this paper,a transport diffuse interface model, which is able to treat zero/unity volume fractions, is presented for solid-fluid interaction. In the proposed model, a new formulation for phase densities is derived, which is unrelated to volume fractions. Consequently, the new model is able to handle zero/unity volume fractions, and nonphysical waves caused by artificial volume fractions are prevented. One-dimensional and two-dimensional numerical tests demonstrate that more accurate results can be obtained by the proposed model.
基金Project (51101059) supported by the National Natural Science Foundation of ChinaProject (20110490874) supported by the China Postdoctoral Science Foundation
文摘A novel approach based on the quantitative phase field model was proposed to calculate the interface mobility and applied to the α/β interface of a ternary Ti-6Al-4V alloy.Phase field simulations indicate that the higher interface mobility leads to the faster transformation rate,but only a unique value of interface mobility matches the diffusion equation under the diffusion-controlled condition.By comparing the transformation kinetics from phase field simulations with that from classical diffusion equation,the interface mobility at different temperatures can be obtained.The results show that the calculated interface mobility increases with increasing temperature and accords with Arrhenius equation very well.
基金the National Natural Science Foundation of China financially(Grant No.10176009)
文摘A new phase field method for two-dimensional simulations of binary alloy solidification was studied. A model basing on solute conservative in every unit was developed for solving the solute diffusion equation during solidification. Two-dimensional computations were performed for ideal solutions and Ni-Cu dendritic growth into an isothermal and highly supersaturated liquid phase.
基金The work is supported by the National Natural Science Foundation of China(No.11401467)China Postdoctoral Science Foundation(No.2013M542334.and No.2015T81012)Natural Science Foundation of Shaanxi Province(No.2015JQ1012).The work is also supported in part by funding from King Abdullah University of Science and Technology(KAUST)through the grant BAS/1/1351-01-01.
文摘In this paper,we study linearly first and second order in time,uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel"scalar auxiliary variable"(SAV)approach,a new developed efficient and accurate method for a large class of gradient flows.The schemes are based on the first order Euler method and the second order backward differential formulas(BDF2)for time discretization,and finite element methods for space discretization.The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps.Various numerical experiments are presented to validate the stability and accuracy of the proposed schemes.
文摘We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification.Here we focus on Stefan problems,and their quasi-static variants,with applications to crystal growth.New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.
基金Project supported by the National Natural Science Foundation of China(Grant No.51571122)the Fundamental Research Funds for the Central UniversitiesChina(Grant No.30920130121012)
文摘The temporal interface microstructures and diffusions in the diffusion couples with the mutual interactions of the temperature gradient, concentration difference and initial aging time of the alloys are studied by phase-field simulation, and the diffusion couples are produced by the initial aged spinodal alloys with different compositions. Temporal composition evolution and volume fraction of the separated phase indicate the element diffusion direction through the interface under the temperature gradient. The increased temperature gradient induces a wide single-phase region on two sides of the interface.The uphill diffusion proceeds through the interface, no matter whether the diffusion direction is up or down with respect to the temperature gradient. For an alloy with short initial aging time, phase transformation accompanying the interdiffusion results in the straight interface with the single-phase regions on both sides. Compared with the temperature gradient,composition difference of diffusion couple and initial aging time of the alloy show greater effects on diffusion and interface microstructure.
基金supported by the National Natural Science Foundation of China (Grant No 50401013)Doctorate Foundation of Northwestern Polytechnical University,China
文摘This paper applies a phase field model for polycrystalline solidification in binary alloys to simulate the formation and growth of the columnar dendritic array under the isothermal and constant cooling conditions. The solidification process and microsegregation in the mushy zone are analysed in detail. It is shown that under the isothermal condition solidification will stop after the formation of the mushy zone, but dendritic coarsening will progress continuously, which results in the decrease of the total interface area. Under the constant cooling condition the mushy zone will solidify and coarsen simultaneously. For the constant cooling solidification, microsegregation predicted by a modified Brody- Flemings model is compared with the simulation results. It is found that the Fourier number which characterizes microsegregation is different for regions with different microstructures. Dendritic coarsening and the larger area of interface should account for the enhanced Fourier number in the region with well developed second dendritic arms.
文摘The phase-field method has emerged as the method of choice for the description of microstructure evolution and phase transitions in metallic materials.Following general thermodynamic laws a set of evolution equations for the structural variables of the system,the so called phase-fields,are derived.The paper reviews shortly the theoretical background of the multi-phase-field.Different examples demonstrating the applicability of the method to technical steels will be presented ranging from deformation of the dendritic strand shell during peritectic transformation,grain growth in Austenite to stress driven growth of Pearlite.
文摘The effects of rotation and gravity on an electro-magneto-thermoelastic medium with diffusion and voids in a generalized thermoplastic half-space are studied by using the Lord-Shulman (L-S) model and the dual-phase-lag (DPL) model. The analytical solutions for the displacements, stresses, temperature, diffusion concentration, and volume fraction field with different values of the magnetic field, the rotation, the gravity, and the initial stress are obtained and portrayed graphically. The results indicate that the effects of gravity, rotation, voids, diffusion, initial stress, and electromagnetic field are very pronounced on the physical properties of the material.
基金This work was supported by the National Science Center,Poland(Narodowe Centrum Nauki,Polska)in the project“Statistical modeling of turbulent two-fluid flows with interfaces”(Grant No.2016/21/B/ST8/01010,ID:334165).
文摘This paper reviews differences between the deterministic(sharp and diffuse)and statistical models of the interphase region between the two-phases.In the literature this region is usually referred to as the(macroscopic)interface.Therein,the mesoscopic interface that is defined at the molecular level and agitated by the thermal fluctuations is found with nonzero probability.For this reason,in this work,the interphase region is called the mesoscopic intermittency/transition region.To this purpose,the first part of the present work gives the rationale for introduction of the mesoscopic intermittency region statistical model.It is argued that classical(deterministic)sharp and diffuse models do not explain the experimental and numerical results presented in the literature.Afterwards,it is elucidated that a statistical model of the mesoscopic intermittency region(SMIR)combines existing sharp and diffuse models into a single coherent framework and explains published experimental and numerical results.In the second part of the present paper,the SMIR is used for the first time to predict equilibrium and nonequilibrium two-phase flow in the numerical simulation.To this goal,a two-dimensional rising gas bubble is studied;obtained numerical results are used as a basis to discuss differences between the deterministic and statistical models showing the statistical description has a potential to account for the physical phenomena not previously considered in the computer simulations.
基金supported in part by US NSF-DMS 1016073,NSFC 11271350 and 91130019Special Research Funds for State Key Laboratories Y22612A33S+1 种基金China 863 project 2010AA012301 and 2012AA01A304China 973 project 2011CB309702.
文摘We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gradient recovery techniques,we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters.Numerical examples and applications to the topological analysis of point clouds are also presented.
