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.

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.

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.

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 modeling of dendritic growth of magnesium alloys with a parallel-adaptive mesh refinement algorithm

A two-dimensional phase field(PF)model was developed to simulate the dendritic solidification in magnesium alloy with hcp crystal structure.By applying a parallel-adaptive mesh refinement(Para-AMR)algorithm,the computational efficiency of the numerical model was greatly improved.Based on the PF model,a series of simulation cases were conducted and the results showed that the anisotropy coefficient and coupling coefficient had a great influence on the dendritic morphology of magnesium alloy.The dendritic growth kinetics was determined by the undercooling and equilibrium solute partition coefficient.A significant finding is acquired that with a large undercooling,the maximum solute concentration is located on both sides of the dendrite tip in the liquid,whereas the maximum solute concentration gradient is located right ahead of the dendrite tip in the liquid.The dendrite tip growth velocity decreases with the increase of the equilibrium solute partition coefficient,while the variation trend of the dendrite tip radius is the opposite.Quantitative analysis was carried out relating to the dendritic morphology and growth kinetics,and the simulated results are consistent with the theoretical models proposed in the previously published works.

Effect of grain boundary energy anisotropy on grain growth in ZK60 alloy using a 3D phase-field modeling

A three-dimensional(3D)multiple phase field model,which takes into account the grain boundary(GB)energy anisotropy caused by texture,is established based on real grain orientations and Read-Shockley model.The model is applied to the grain growth process of polycrystalline Mg(ZK60)alloy to investigate the evolution characteristics in different systems with varying proportions of low-angle grain boundary(LAGB)caused by different texture levels.It is found that the GB energy anisotropy can cause the grain growth kinetics to change,namely,higher texture levels(also means higher LAGB proportion)result in lower kinetics,and vice versa.The simulation results also show that the topological characteristics,such as LAGB proportion and distribution of grain size,undergo different evolution characteristics in different systems,and a more serious grain size fluctuation can be caused by a higher texture level.The mechanism is mainly the slower evolution of textured grains in their accumulation area and the faster coarsening rate of non-textured grains.Therefore,weakening the texture level is an effective way for implementing a desired homogenized microstructure in ZK60 Mg alloy.The rules revealed by the simulation results should be of great significance for revealing how the GB anisotropy affects the evolution of polycrystalline during the grain growth after recrystallization and offer the ideas for processing the alloy and optimizing the microstructure.

Numerical Simulation of Two-Dimensional Dendritic Growth Using Phase-Field Model

In this article, we study the phase-field model of solidification for numerical simulation of dendritic crystal growth that occurs during the casting of metals and alloys. Phase-field model of solidification describes the physics of dendritic growth in any material during the process of under cooling. The numerical procedure in this work is based on finite difference scheme for space and the 4th-order Runge-Kutta method for time discretization. The effect of each physical parameter on the shape and growth of dendritic crystal is studied and visualized in detail.

Microstructural evolution and microsegregation in directional solidification of hypoeutectic Al−Cu alloy:A comparison between experimental data and numerical results obtained via phase-field model

The objective of the work is focused on predictions of microsegregation,solidification speed,dendritic arm spacings and dendrite morphology by phase-field model.The numerical results were compared with experimental data.The experimental values for cooling rates and effective partition coefficient were adopted during calculations.The results of microsegregation through phase-field model show excellent agreement with the experimental data.Such excellent agreement is because cooling rates,effective partition coefficient and back-diffusion of solute are considered in the model.For solidification speed,the calculation results show good agreement with the experimental data.Tertiary dendritic arm spacing calculated with phase-field model is compared with experimental data.The results show good agreement between them.The dendrite arm spacing varies with position because high cooling rates are responsible for the refinement effect on microstructure.Finally,two-dimensional simulation produced a dendrite that is similar to that found in the experiment.

Phase-field modeling of faceted growth in solidification of alloys

A regularization of the surface tension anisotropic function used in vapor-liquid-solid nanowire growth was introduced into the quantitative phase-field model to simulate the faceted growth in solidification of alloys.Predicted results show that the value of δ can only affect the region near the tip,and the convergence with respect to δ can be achieved with the decrease of δ near the tip.It can be found that the steady growth velocity is not a mo no tonic function of the cusp amplitude,and the maximum value is approximately at ε=0.8 when the supersaturation is fixed.Moreover,the growth velocity is an increasing function of supersaturation with the morphological transition from facet to dendrite.

Global weak solutions to a phase-field model for motion of grain boundaries

We employ the Galerkin method to prove the global existence of weak solutions to a phase-field model which is suitable to describe a sort of interface motion driven by configurational forces.The higher-order derivative of unknown S exists in the sense of local weak derivatives since it may be not summable over the original open domain.The existence proof is valid in the one-dimensional case.

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 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.

Growth and inhibition of zinc anode dendrites in Zn-air batteries:Model and experiment

Zinc(Zn)-air batteries are widely used in secondary battery research owing to their high theoretical energy density,good electrochemical reversibility,stable discharge performance,and low cost of the anode active material Zn.However,the Zn anode also leads to many challenges,including dendrite growth,deformation,and hydrogen precipitation self-corrosion.In this context,Zn dendrite growth has a greater impact on the cycle lives.In this dissertation,a dendrite growth model for a Zn-air battery was established based on electrochemical phase field theory,and the effects of the charging time,anisotropy strength,and electrolyte temperature on the morphology and growth height of Zn dendrites were studied.A series of experiments was designed with different gradient influencing factors in subsequent experiments to verify the theoretical simulations,including elevated electrolyte temperatures,flowing electrolytes,and pulsed charging.The simulation results show that the growth of Zn dendrites is controlled mainly by diffusion and mass transfer processes,whereas the electrolyte temperature,flow rate,and interfacial energy anisotropy intensity are the main factors.The experimental results show that an optimal electrolyte temperature of 343.15 K,an optimal electrolyte flow rate of 40 ml·min^(-1),and an effective pulse charging mode.

Allen-Cahn型相场模型的移动网格方法

针对2种流体的Allen-Cahn型相场模型提出了一种自适应移动网格方法。移动网格方法包括网格重分布和偏微分方程求解2个相对独立的部分。通过求解一个类似于Poisson方程的偏微分方程组获得网格重分布,大量的网格点聚集在2种流体的界面附近从而提高了分辨率,而在其他的区域则仅有比较稀疏的网格点。Allen-Cahn模型用有限差分方法求解,其中的大型稀疏线性方程组使用代数多重网格快速算法求解。数值实验表明移动网格算法在提高分辨率和计算效率方面非常有效。

Convection effect on dendritic growth using phase-field method

Phase-field model was employed to quantitatively study the effect of convection on pattern selection and growth rate of 2D and 3D dendrite tip,as well as the effect of the different convection velocity on the dendritic growth.The calculated results show that crystal is asymmetric in the priority direction of growth under flow.The dentritic growth is promoted in the upstream region and suppressed in the downstream region.Convection can cause deviation in the dendrite growth direction and the preferred direction of the columnar crystals.It has been found that both primary dendrite stem and secondary dendrite arm deflect significantly towards upstream direction,secondary dendrite arm in upstream direction is more developed than the primary dendrite in downstream direction.

Numerical Simulation of Solidification Microstructure and Effects of Phase-field Parameters on Grain Growth Morphologies

By a simple phase field model, a series of numerical simulations of solidification microstructure was performed to show a rich variety of dendritic patterns. At the same time, the relation between the morphology of grain growth and some parameters including the strength of anisotropy, dimensionless latent heat and the size of initial solid zone was studied. It is for the first time that patterns of grain growth were associated with the size of initial solid zone, which is an interesting issue. The possible reason for this may be that variation in the size of initial solid zone may bring about fluctuation of the interface energy, making the interface unstable.

Phase-field simulation of dendritic solidification using a full threaded tree with adaptive meshing

Simulation of the microstructure evolution during solidifi cation is greatly benefi cial to the control of solidifi cation microstructures. A phase-fi eld method based on the full threaded tree(FTT) for the simulation of casting solidifi cation microstructure was proposed in this paper, and the structure of the full threaded tree and the mesh refi nement method was discussed. During dendritic growth in solidifi cation, the mesh for simulation is adaptively refi ned at the liquid-solid interface, and coarsened in other areas. The numerical results of a threedimension dendrite growth indicate that the phase-fi eld method based on FTT is suitable for microstructure simulation. Most importantly, the FTT method can increase the spatial and temporal resolutions beyond the limits imposed by the available hardware compared with the conventional uniform mesh. At the simulation time of 0.03 s in this study, the computer memory used for computation is no more than 10 MB with the FTT method, while it is about 50 MB with the uniform mesh method. In addition, the proposed FTT method is more effi cient in computation time when compared with the uniform mesh method. It would take about 20 h for the uniform mesh method, while only 2 h for the FTT method for computation when the solidifi cation time is 0.17 s in this study.

Microstructure analyses and phase-field simulation of partially divorced eutectic solidification in hypoeutectic Mg-Al Alloys

In this study the partially divorced eutectic microstructure ofα-Mg andβ-Mg17Al12was investigated by electron backscatter diffraction,transmission electron microscopy,and phase-field modeling in hypoeutectic Mg-Al alloys.The orientation relationships between the individual eutecticαgrains,eutecticβphase,and primaryαgrains were investigated.While the amount of eutectic morphology is primarily determined by the Al content,the in-depth microstructure analyses and the phase-field simulation suggest non-interactive nucleation and growth of eutecticαphase in theβphase grown on the interdendritic primaryαdendrites.Also,phase-field simulations showed a preferred nucleation sequence where theβphase nucleates first and subsequently triggers the nucleation of eutecticαphase at the movingβphase solidification front,which supports the microstructural analysis results.

Phase-field simulation of dendritic growth in a binary alloy with thermodynamics data

This paper simulates the dendrite growth process during non-isothermal solidification in the Al-Cu binary alloy by using the phase-field model. The heat transfer equation is solved simultaneously. The thermodynamic and kinetic parameters are directly obtained from existing database by using the Calculation of Phase Diagram （CALPHAD） method. The effects of the latent heat and undercooling on the dendrite growth, solute and temperature profile during the solidification of binary alloy are investigated. The results indicate that the dendrite growing morphologies could be simulated realistically by linking the phase-field method to CALPHAD. The secondary arms of solidification dendritic are better developed with the increase of undercooling. Correspondingly, the tip speed and the solute segregation in solid-liquid interface increase, but the tip radius decreases.