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.

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.

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.

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.

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.

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。

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.

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.

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.

Multiphase-field simulation of austenite reversion in medium-Mn steels

Medium-Mn steels have attracted immense attention for automotive applications owing to their outstanding combination of high strength and superior ductility.This steel class is generally characterized by an ultrafine-grained duplex microstructure consisting of ferrite and a large amount of austenite.Such a unique microstructure is processed by intercritical annealing,where austenite reversion occurs in a fine martensitic matrix.In the present study,austenite reversion in a medium-Mn alloy was simulated by the multiphase-field approach using the commercial software MICRESS®coupled with the thermodynamic database TCFE8 and the kinetic database MOBFE2.In particular,a faceted anisotropy model was incorporated to replicate the lamellar morphology of reversed austenite.The simulated microstructural morphology and phase transformation kinetics(indicated by the amount of phase)concurred well with experimental observations by scanning electron microscopy and in situ synchrotron high-energy X-ray diffraction,respectively.

Three-dimensional multi-phase-field simulation of eutectoid alloy based on OpenCL parallel

Seeking high-performance computing methods to solve the problem of a large amount of calculation,low calculation efficiency,and small simulation scale on the traditional single central processing unit (CPU) platform is of great value to the simulation study of micro-structure.In this study,based on the three-dimensional multi-phase-field model of KKSO coupling phase-field and solute field,the open computing language (OpenCL) + graphics processing unit (GPU) heterogeneous parallel computing technology is used to simulate the eutectoid growth of Fe-C alloy and the end growth process of pearlite under pure diffusion.The effects of initial supercooling and different diffusion coefficients on the growth morphology of lamellar pearlite were investigated.The results show that ferrite and cementite are perpendicular to the front of the solid-solid interface and are coupled and coordinated to grow,and there is no leading phase under the initial supercooling degree of 20 K.With the continuous increase of the initial supercooling degree (19 K-22 K),the morphology changes of the eutectoid layer are as follows:cementite stops growing → slice amplitude increases → regular symmetric growth → oblique growth → layer merge.With the increase of the diffusion coefficient from 3×10^(-13) m^(2)·s^(-1) to 15×10^(-13) m^(2)·s^(-1),the growth rate of the microstructure of the lamellar pearlite increases linearly,and there is no obvious change in the frontal appearance of the pearlite.

Microscopic phase-field simulation of L1_2 and D0_(22) phases during the early precipitation process of Ni-Cr-Al alloys

The influence of temperature on the precipitation mechanism and sequence of L 12 and D022 phases during the early precipitation process of a Ni-15.Sat%Cr-14at%Al alloy was simulated based on the microscopic phase-field model. In the range from 873 to 1373 K, the precipitation mechanism transformed from spinodal decomposition to non-classic nucleation and growth; the incubation period prolonged gradually with increasing temperature. The volume fraction of L12 phases increased and that of D022 phases decreased. D022 phases disappeared at 1373 K, and finally single-phase L12 phases were formed.

Microscopic Phase-field Simulation of Competition Mechanism Between L12 and D022 Structure in Ni-Cr-Al Alloy

Simulations are performed on temporal evolution of atom morphology and ordering parameters of Ni-14.5 Cr-16.5 Al alloy during early precipitation process at different temperatures based on microscopic phase-field theory; the relationship between precipitation sequence and mechanism of L12 and D022 structure and precipitation temperature are illuminated. The nonstoichiometric ordered L12 phases appear first with congruent ordering＋spinodal decomposition mechanism which is then followed by precipitation of D022 phases at ordering domain boundaries of L12 phases by spinodal decomposition mechanism at 1073 K and 1223 K. The nonstoichiometric L12 phases transform to stoichiometric ordering phases gradually. The incubation period of L12 and D022 phases is shorter at 1073 K than that 1223 K, and growth speed is higher at 1073 K. At 1373 K, L12 and D022 phases appear simultaneously by non-classical nucleation and growth mechanism. After that the particles of D022 phases diminish and disappear gradually; L12 phases grow and single L12 phases are remained at last.