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The macroscopic hysteresis of ferromagnetic materials is influenced by their microscopic magnetic domains, establishing a correspondence between the two that enables a fundamental understanding of the magnetization process of ferromagnetic materials. Based on this, this study employs the phase field method to investigate the macroscopic hysteresis and microscopic magnetic domain evolution of soft, hard, and rectangular ferromagnetic materials. The hysteresis loops of the three materials present narrow, wide and rectangular characteristics respectively, which are quantitatively consistent with the experimental data. Subsequently, the influence of the magnetic field on the macroscopic hysteresis is analyzed. The results show that the hysteresis loop area increases when the magnetic field amplitude increases. Furthermore, the proportion of each type of energy under different magnetic fields is discussed, the dominant energy terms are generally consistent for the same category of ferromagnetic materials. Additionally, the corresponding magnetic domain evolution under different magnetic field is displayed, with soft magnetic materials exhibiting no vortex structures, hard ferromagnetic materials presenting a 45° vortex structure, and rectangular magnetic material showcasing a 90° vortex structure. Finally, the asymmetric characteristics of the major hysteresis loop and minor hysteresis loops are discussed. The change of the magnetic field path leads to the corresponding change of the major and minor hysteresis loops. At the same magnetic field, distinctions in magnetic domain configurations between the major and minor hysteresis loops are evident.

The laminated sandwich composites have wide structure-making applications in the automotive and aviation fields due to their lightweight and superior flexural rigidity properties. Making grooves or holes to assemble more than one structure induces crack discontinuities near the stress concentration region in these sandwich structures. The present work examines the effect of crack discontinuities on the mechanical performance and failure process of the sandwich structures under different loading conditions. Phase field method (PFM) has been presented and implemented using in-house developed MATLAB code. The effect of holes, multiple cracks, number of cores, and loading conditions are analyzed for the mechanical and fracture behavior of the structure. Load-carrying capacity, threshold displacement value for crack initiation, crack propagation trajectory, and energy absorption capacity are compared for various crack discontinuities under different loading conditions. Approximately 35% increase in load carrying capacity is observed in equivalent multiple core sandwich structures.

This paper establishes a microchannel model based on the pore scale distribution characteristics of natural cores from Daqing Oilfield. It considers the deformation and flow characteristics of the dispersed and continuous phases in the heterogeneous system， builds a flow model by the phase field method， and solves it by the finite element method. The paper also simulates the generation of dispersed phase particles in the microchannel， realizes particle sorting， and studies the effect of the matching coefficient and pore-throat ratio on the plugging performance of particles in the microscopic pore-throat structure. The results show that when the particles are elastically plugged in the microscopic pore-throat structure， the pressure at the entrance of the pore throat changes periodically with the migration of the particles through the pore throat. The optimal matching coefficient between the particles and the pore throat is ［1.0， 1.4）. In this interval， the particles can be temporarily plugged at the entrance of the pore throat and recover their original shape after deformation and migration through the pore throat. When the pore diameter is the same， a larger matching coefficient and pore-throat ratio indicate greater pressure of particles through the pore throat， and larger particle size reflects a smaller critical value of particles through the pore throat.

Acid fracturing is a crucial stimulation technique to enhance hydrocarbon recovery in carbonate reservoirs. However, the interaction between acid fractures and natural fractures remains complex due to the combined effects of mechanical, chemical, and fluid flow processes. This study extends a previously developed hydro-mechano-reactive flow coupled model to analyze these interactions, focusing on the influence of acid dissolution. The model incorporates reservoir heterogeneity and simulates various scenarios, including different stress differences, approaching angles, injection rates, and acid concentrations. Numerical simulations reveal distinct propagation modes for acid and hydraulic fractures, highlighting the significant influence of acid dissolution on fracture behavior. Results show that hydraulic fractures are more likely to cross natural fractures, whereas acid fractures tend to be arrested due to wormhole formation. Increasing stress differences and approaching angles promote fracture crossing, while lower angles favor diversion into natural fractures. Higher injection rates facilitate fracture crossing by increasing pressure accumulation, but excessive acid concentrations hinder fracture initiation due to enhanced wormhole formation. The study demonstrates the importance of tailoring fracturing treatments to specific reservoir conditions, optimizing parameters to enhance fracture propagation and reservoir stimulation. These findings contribute to a deeper understanding of fracture mechanics in heterogeneous reservoirs and offer practical implications for improving the efficiency of hydraulic fracturing operations in unconventional reservoirs.

In recent decades, the phase field method has been widely used in order to model and simulate the damage in various materials and/ or structures. In this simulation method, the regularization length is an important parameter to describe the width of the smeared crack and reflect the crack as a sharp discontinuity. The regularization parameter depends on the material properties thus its value must be small enough. This leads to the element mesh size being small, in other words, the number of elements increases, causing computation costs to much increase. On the other hand, in brittle materials, the positive and negative parts of the strain tensor represent the tension and compression behaviours in the materials. Two parts of the strain tensor must satisfy strain orthogonal decompositions in the context of the elastic stiffness tensor behaving as a metric. Therefore, in this work, the phase field method is incorporated into the improved degradation functions and strain orthogonal condition in order to investigate the crack nucleation and propagation as well as predict the peak load and/ or the critical stress corresponding to the first crack onset appeared in the experimental brittle material such as plaster. A comparison between the obtained results and results of the available experimental tests and/ or relevant simulation methods will demonstrate that the present proposed method makes the mesh size coarser thus the computational cost is significantly reduced without changing the crack path. Moreover, the present simulation method helps to raise the accuracy of the global and local mechanical responses in the material, which is represented by smoother relationship curves.

Abstract Damage and fracture are the most extensive failure modes of rock materials, which may easily induce disaster and instability of engineering structures. This study developed a nonlocal damage fracture phase field model for rocks considering the heterogeneity of rocks. The modified phase field model introduced the heterogeneity of fracture parameters and modified the governing equations. Meanwhile, the free energy was constructed by the elastic strain energy sphere‐bias decomposition and the plastic strain energy. As for the numerical implementation, the three layers finite elements method structure was used in the frame of the finite element method. The ability of the modified phase field model has been illustrated by reproducing the experiment results of rock samples with pre‐existing cracks under compression.

Electron beam melting is a powder bed fusion process capable of manufacturing thin structural features. However, as the thickness of these features approaches typical microstructure grain sizes, it becomes vital to understand how the manufacturing process contributes to local crystallographic texture and anisotropy in micromechanical response. Therefore, this article investigates Ti–6Al–4V ⍺/β-phase formation within thin components using a variety of experimental and numerical approaches. Optical and scanning electron microscopy are used to determine through-thickness distributions of prior-β width ([top, middle, bottom]:[81.2 ± 44.2, 76.02 ± 30.4, 75.6 ± 31.2] μm), ⍺-lath thickness ([top, middle, bottom]:[1.0 ± 1.3, 1.3 ± 1.2, 1.4 ± 1.8] μm; average), and ⍺/β-phase fractions ([top, middle, bottom]:[0.87 ± 0.05, 0.82 ± 0.03, 0.88 ± 0.03]; average). Manufacturing process (i.e., “logfile”) data is used within a layer-by-layer finite element “birth/death” model. This model is loosely coupled with the Kim-Kim-Suzuki phase field model and a CALPHAD thermodynamic database to predict ⍺-lath growth throughout the process. In general, good correlation is found between the experimental data and the predicted temperature history, ⍺-lath coarsening, and phase fraction. This indicates that these tools would be useful in predicting process-structure-properties-performance relationships for thin features.

Advanced damage tolerance design of materials and mechanical components heavily relies on fracture failure analysis. A robust, efficient, and versatile software (GPFniCS) is developed and provided for public access to perform fracture analyses based on the Generalised Phase-Field Method. GPFniCS software is developed on top of FEniCS, an open-source finite element library. One-dimensional and two-dimensional mixed mode problems are validated with GPFniCS and provided as illustrative examples in a public repository. The software shows excellent potential for computational fracture studies, and it is open to further developments in various fields like thermal loading, fatigue loading, solidification, and many more.

The phase field model - a powerful tool - has been well established to simulate the fatigue crack evolution behavior. However, it is still hard to understand how each energy component in the phase field model contributes to crack evolution since the phase field method is based on an energetic criterion. In this work, we borrow the concept of configurational forces and show a straightforward way to examine the energetic driving forces in the phase field fatigue model. Results show that different parts of the configurational forces provide different energetic contributions during crack propagation.

A specific columnar crystal structure is obtained using the directional solidification technique, which has a substantial effect on the optimization of the axial mechanical properties of the alloy. Additionally, the convection phenomenon in the melt changes the temperature field and concentration field at the front of the solid–liquid interface, affecting the shape of this interface. Thus, the influence on alloy properties cannot be ignored. Although the phase field method has more research on the microdendrite growth morphology, the results of coupling the flow field into the phase field and exploring the microdendrite morphology of directional solidification are still scarce. In this paper, the phase field model of a coupled flow field is used to simulate dendritic growth during directional solidification. The effects of the anisotropy coefficient and interfacial energy on the growth of directionally solidified dendrites and the growth behavior of dendrites under forced convection were studied. For the numerical solution procedure, a uniform grid of the finite difference method was used to discretize the governing equations. A combined solution of the MAC algorithm and a phase field discrete calculation was realized. When addressing the coupling of the microvelocity and pressure fields, the MAC algorithm was used to solve the Navier–Stokes equation and pressure Poisson equation, and the interlocked grid method was applied to handle the complex free interface. The results show that the growth rate of the dendrite tip increases, and the radius of curvature and the solute concentration at the root of the dendrite decrease with an increasing anisotropy coefficient. When the anisotropy coefficient is a maximum of 0.065, the wall of the dendrite tends to develop toward a secondary dendrite because of the influence of the anisotropy coefficient; with increasing interfacial energy, the radius of curvature of the dendrite tip increases. When the interfacial energy is a maximum of 0.6 J·m−2, the solidification shows a flat interface advancing mode; forced convection has a great influence on the growth direction of directional solidification dendrites. The directional solidification of dendrites in the upstream direction is coarse and grows faster with increasing flow rate. Additionally, the dendrite growth morphology observed using an optical microscope agrees well with the experimental results.

The solutal convection is able to affect the dendrite growth and solute distribution during directional solidification. The solutal convection is raised by the solute difference-caused buoyancy. When solutal convection becomes intensive, solute plumes will occur, which may cause freckle defects in directional solidification of superalloys. The onset of solutal plumes depends on the solidification condition. In this study, the dendrite growth with solutal convection during directional solidification of nickel-based superalloy was investigated by two-dimensional phase-field simulations. The lattice Boltzmann method was used to solve the flow field driven by solute difference caused-buoyancy force. The pseudo-binary alloy approximation was adopted for simplification of the multicomponent alloy. The effects of the temperature gradient, pulling velocity, and isotherm inclination angle on dendrite growth and solutal convection were investigated. The dendrite growth dynamics and solutal convection pattern under different solidification conditions were analyzed. Dendrite growth velocity oscillation was observed. The effect of pulling velocity on dendrite growth velocity oscillation was analyzed.

To explore the demulsification mechanism of emulsion under electric fields, the movement process of dispersed phase droplets in water-in-oil emulsion was simulated with the finite element method. Herein, the oil-water distribution was represented with the phase field method, and the accuracy and efficiency of calculation on fluid flow process was improved with the structured quadrilateral grid. Besides, the influence laws of electric field parameters, as well as the physical parameters of emulsion, on the droplet deformation and coalescence were systematically studied to provide theoretical basis for the efficient application of electric demulsification technology. According to the results, the higher the electric field intensity, root mean square(RMS) of electric field waveform and droplet size, the greater the deformability and coalescence ability of droplets are. The droplet deformation is independent of frequency under the high-frequency conditions, and the frequency has little effect on the coalescence rate of droplets. Meanwhile, the droplet deformation and coalescence rate can been improved by decreasing the continuous phase viscosity and oil-water interfacial tension. In addition, the microcosmic mechanism of droplet deformation and coalescence was also discussed. It is found that the maximum droplet deformation is in strong linear relationship with the electric Weber number in the case of small deformation, and the electric field intensity,droplet size and interfacial tension affect the coalescence efficiency of droplets by influencing the evolution of liquid bridge between droplets.

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

Objective: In order to explore the effect of an external electrostatic field on the growth of ice crystals during food freezing. Methods: Based on the classic Kobayashi model of phase field method, the free energy density of electric field was introduced into the phase field model. Used the five-point difference scheme in the finite difference method to discretize the partial differential equation, the simulation and visualization of ice crystal growth were realized by programming. Results: The results showed that the simulation results were consistent with other simulation results and experimental results, and the ice crystal morphology had a typical sixfold symmetry; With the increase of electric field intensity, the growth rate of main branches and nucleation growth of ice crystals were inhibited. Conclusion: In the process of quick-frozen food preservation, adding a certain electric field strength will reduce the formation of ice crystals, and retain the quality and flavor of food as much as possible.

The paper presents a framework for accelerating the phase field modeling of compressive failure of rocks. In this study, the Drucker-Prager failure surface is taken into account in the phase field model to characterize the tension-compression asymmetry of fractures in rocks. The degradation function that decouples the phase-field and physical length scales is employed, in order to reduce the mesh density in large structures. To evaluate the proposed approach, four numerical examples are given. The results of the numerical experiments demonstrate the accuracy and efficiency of the proposed approach in tracking crack propagation paths in rock materials under Drucker-Prager criterion.

Abstract Ultra pure metals have various applications, e. g. as electrical conductors. Crystallization from the melt, e. g. via zone melting, using the segregation of impurities at the solidification front is the basic mechanism behind different technical processes for the refining of metals and semi-metals. In this paper, we focus on a crystallization methodology with a gas cooled tube (“cooled finger”) dipped into a metallic melt in a rotating crucible. The necessary requirement for purification in a solidification process is a morphologically stable solidification front. This is the only way to enable macroscopic separation of the impurities, e. g. by convection. For cellular or dendritic solidification morphologies, the segregated impurities are trapped into the interdendritic melt and remain as microsegregations in the solidified metal. Morphological stability depends on the temperature gradient G at the solidification front, the solidification front velocity V front and thermodynamic alloy properties like the segregation coefficients of the impurity elements. To quantify the impact of these parameters on the morphological evolution, especially on the planar/cellular transition and thus on microsegregation profiles, phase field simulations coupled to a thermodynamic database are performed for an aluminium melt with three impurities, Si, Mn and Fe. In particular, we have investigated the morphology evolution from the start of solidification at the cooled finger towards a stationary growth regime, because in the technical process a significant fraction of the melt solidifies along the initial transient. To solve the transient long range temperature evolution on an experimental length scale, the temperature field has been calculated using the homoenthalpic approach together with a 1D temperature field approximation. The simulations provide the process window for an energy efficient purification process, i. e. low thermal gradients, and elucidate the benefit of melt convection.

Fracture is one of the most common failure modes of materials and components and greatly restricts engineering design. Understanding of the crack propagation and evolution of rock and other engineering materials is of great significance to engineering construction. For the current numerical methods there are more or less limitations when analyzing the evolution of cracks, such as the mesh dependence of the crack path, the difficulty to deal with crack bifurcation and merging by the classic fracture criterion. In recent years, the phase field method (PFM) has been widely used in simulating crack growth. A phase field numerical manifold method (PFNMM) makes use of the advantages of the phase field method in simulating crack propagation and those of the numerical manifold method (NMM), is proposed for crack growth in rock. The implementation details of the proposed numerical model are presented. Several benchmark examples, including notched semi-circular bend test and Brazilian disc test, are adopted to validate the proposed numerical approach. After that, the multi-crack propagation process with different rock bridge inclination angles under uniaxial compression is simulated, which is in good agreement with the results derived from laboratory and PFC. And the results indicate that the PFNMM has broad application prospects in simulating crack growth of rock.

Reinforced concrete has been widely used. However, it is difficult to accurately simulate the process of concrete damage and destruction caused by rebar corrosion. Therefore, the objective of this work is to solve this difficult problem using a phase field method and analyze the failure process of reinforced concrete. The main importance of this work is to improve the calculation method for the problem, simulate the damage and destruction of concrete cover, and visualize the crack propagation process. Based on the strain softening characteristics of concrete the damage of concrete cover caused by uniform and non-uniform corrosion of rebars is simulated respectively. The simulation results are consistent with the actual observation. The applicability of the phase field method for complex crack propagation in reinforced concrete is verified, which provides a reference for the engineering design and durability research of structures. Analysis of simulation results found that the crack propagation speed is fast for uniform corrosion. Increasing the thickness of concrete cover is benefit to improve the corrosion resistance of structure. However, the crack propagation speed is slow for non-uniform corrosion. The thickness of the cover has no obvious effect on the corrosion damage resistance of structure. For the non-uniform corrosion of multiple rebars, depending on the thickness of concrete cover and the spacing between rebars, the concrete cover presents three typical crack propagation morphologies: wedge-shaped cracks, layered cracks and small local cracks.

Abstract In topology optimization the goal is to find the ideal material distribution in a domain subject to external forces. The structure is optimal if it has the highest possible stiffness. A volume constraint ensures filigree structures, which are regulated via a Ginzburg–Landau term. During 3D printing overhangs lead to instabilities. As a remedy an additive manufacturing constraint is added to the cost functional. First order optimality conditions are derived using a formal Lagrangian approach. With an Allen-Cahn interface propagation the optimization problem is solved iteratively. At a low computational cost the additive manufacturing constraint brings about support structures, which can be fine tuned according to demands and increase stability during the printing process.

Based on the Kim-Kim-Suzuki (KKS) phase field model coupled with the thermodynamic parameters, the transformation process from columnar dendrites to equiaxed crystals during directional solidification of aluminium alloy was simulated, and the effects of phase field parameters on the growth morphology and dendrite segregation were discussed. Furthermore, considering the effect of the microcosmic flow field, the convection influence gradient term is introduced into KKS formula near the solid-liquid interface, and the phase field model considering flow field was applied to the inherent convective environment of the actual roll casting process, also the multiple dendrites growth behavior of magnesium alloy under the action of microscopic convection was further explored. When coupling calculation of microscopic velocity field and pressure field, the staggered grid method was used to deal with the complex interface. The combined solution of Marker in Cell (MAC) algorithm and phase field discrete calculation was realized. In order to further describe the influence of convection on the solidification process, the roll casting experiments are used to verify the impact growth of multiple dendrites under convection. The results show that the dendrites undergo solute remelting and the dendrites melt into equiaxed crystals, showing the phenomenon of Columnar to Equiaxed Transition (CET).

A radial well has a unique wellbore configuration. Fracture propagation in radial well deflagration fracturing is studied rarely. The mechanism of interaction between deflagration fractures, natural fractures, and micro-fractures is still unknown. Based on continuum mechanics, damage mechanics, and variational principles, a numerical model of fracture propagation in deflagration fracturing is established with the Hamilton principle and phase-field fracture theory. The effects of horizontal principal stress difference, natural fracture distribution, and micro-fractures around the wellbore on fracture propagation in deflagration fracturing are studied. First, when no natural fractures are developed around the radial well, fractures are initiated at both ends of the radial well. Second, when there are three natural fractures around the radial well, the created fractures have the morphology of shorter fractures in the middle and longer fractures on both sides under stress interference mechanisms. Third, a larger density of natural fractures causes obvious stress superposition, changes the initiation points of radial wells and fracture morphology, and increases fracture width and reservoir stimulation volume. Fourth, as the micro-fractures increase, their interference and induction effects on deflagration fractures are enhanced gradually, and the deflection angle of fractures increases by 38.7%. The study provides a reference for optimizing deflagration fracturing in a radial well.

This paper aims to develop a new numerical model to evaluate the flexural damage behavior of pervious concrete from its compositions. By considering this material within the framework of the quasi elasto-brittle approach, failure is modeled by the phase field method. Then, a taking-placing generation process of the extensive Monte Carlo simulation-type was constructed to account for the statistical effect of the pervious concrete structure. Various numerical investigations have been performed using this model. First, the modeling parameters were calibrated with the three-point bending test, then numerical-experimental correlations were well captured for both elastic and post softening regimes for the different structure sizes. Archived crack pattern analyses were used to examine the influence of the pore structure on certain values of peak load. Finally, the linear relationship between the flexural strength and the porosity was confirmed by agreement with the simulation results, which comprised of empirical references.

In order to study the microcosmic mechanism of ice formation in ship structure, a numerical simulation is conducted. Based on the Wheeler phase field model, the finite difference method is used to simulate the solidification shape of seawater at different undercoolings. In the process of simulation, sea water is regarded as the binary mixture of salt and pure water, and the authenticity of ice crystal growth is ensured by setting ice physical parameters and introducing crystal nucleus. In the calculation, four nuclei are set up to simulate the ice crystal growth, and the influence of different undercoolings on the ice crystal growth is discussed. The results show that the growth rate of ice crystal increases with the increase of undercooling, and it is linear with the change of undercooling. When the dimensionless undercooling is less than 0.85, the dendrite has only a few branches which are thin. When the dimensionless undercooling is higher than 0.85, the ice crystal has more than two-stages of branches which are obviously thicker. When the dimensionless undercooling reaches 1.0, the branches are squeezed by the main branches and lessned, and the crystal cores are finally filled.

As an ideal high-density storage unit, magnetic nanorings have become a research hotspot in recent years. We can both study the evolution of microscopic state of magnetization and acquire macroscopic magnetic properties by micromagnetic simulation, which has thus been widely used. However, traditional micromagnetism cannot simulate complex stress state. Due to the introduction of microelasticity theory, the phase field method for magnetic materials can be used to calculate the coupling effect of stress and magnetic field. However, the computing model usually needs to satisfy periodic boundary condition. In this paper, the phase field simulation combined with the finite element method is employed. By using user defined element, the evolution of magnetic domain structures of the double-hole nanorings has been studied. In different diameter of the holes and external magnetic field direction, we have found seven kinds of magnetic domain evolution mechanism. Among them, the twin-vortex evolution mechanism with high stability and low demagnetization interference characteristics of advantages, has good application prospect in magnetic random-access memory (MRAM) unit.

The distinctions of dendritic morphology and sidebranching behavior when solidified under atmosphere pressure, constant pressure which is higher than atmosphere pressure (hereinafter referred to as constant pressure) and periodic pressure were investigated using 3-D phase field method. When growing at atmosphere pressure, side branches (secondary dendritic arms) are irregular. When solidified under constant pressure with a relatively high value, side branches are much more luxuriant, with more developed high-order side branches. When applied with periodic pressure, resonant sidebranching happens, leading to many more regular side branches and the smallest secondary dendritic arm spacing (SDAS) in the three cases. The significant difference in dendritic morphology is associated with tip velocity modulated by total undercooling including pressure and temperature undercooling. In the case of constant pressure, tip velocity increases linearly with total undercooling, and it varies periodically in periodic pressure case. The different variation trend in tip velocity is the reason for the distinct dendrite growth behavior in different cases. Unlike the phenomenon in constant pressure case where the dendrite grows faster with higher pressure, the dendrite grows slower under periodic pressure with higher amplitude, resulting in less developed primary dendrite and side branches. This is influenced by tip remelting due to low undercooling or even negative undercooling. It is revealed that the accelerated velocity of tip remelting increases with the decline of undercooling. The greater the amplitude of periodic pressure, the faster the tip remelting velocity during one period. This is the reason why the average tip velocity decreases with the rise of amplitude of periodic pressure.

Primary dendrite arm spacing (PDAS) is a crucial microstructural feature in nickel-based superalloys produced by laser cladding. In order to investigate the effects of process parameters on PDAS, a multi-scale model that integrates a 3D transient heat and mass transfer model with a quantitative phase-field model was proposed to simulate the dendritic growth behavior in the molten pool for laser cladding Inconel 718. The values of temperature gradient (G) and solidification rate (R) at the S/L interface of the molten pool under different process conditions were obtained by multi-scale simulation and used as input for the quantitative phase field model. The influence of process parameters on microstructure morphology in the deposition layer was analyzed. The result shows that the dendrite morphology is in good agreement with the experimental result under varying laser power (P) and scanning velocity (V). PDAS was found to be more sensitive to changes in laser scanning velocity, and as the scanning velocity decreased from 12 mm/s to 4 mm/s, the PDAS increased by 197% when the laser power was 1500 W. Furthermore, smaller PDAS can be achieved by combining higher scanning velocity with lower laser power.

In this study, a phase field method based on the Cahn–Hilliard equation was used to simulate the spinodal decomposition in Zr-Nb-Ti alloys, and the effects of Ti concentration and aging temperature (800–925 K) on the spinodal structure of the alloys for 1000 min were investigated. It was found that the spinodal decomposition occurred in the Zr-40Nb-20Ti, Zr-40Nb-25Ti and Zr-33Nb-29Ti alloys aged at 900 K with the formation of the Ti-rich phases and Ti-poor phases. The spinodal phases in the Zr-40Nb-20Ti, Zr-40Nb-25Ti and Zr-33Nb-29Ti alloys aged at 900 K were in an interconnected non-oriented maze-like shape, a discrete droplet-like shape and a clustering sheet-like shape in the early aging period, respectively. With the increase in Ti concentration of the Zr-Nb-Ti alloys, the wavelength of the concentration modulation increased but amplitude decreased. The aging temperature had an important influence on the spinodal decomposition of the Zr-Nb-Ti alloy system. For the Zr-40Nb-25Ti alloy, with the increase in the aging temperature, the shape of the rich Zr phase changed from an interconnected non-oriented maze-like shape to a discrete droplet-like shape, and the wavelength of the concentration modulate increased quickly to a stable value, but the amplitude decreased in the alloy. As the aging temperature increased to 925 K, the spinodal decomposition did not occur in the Zr-40Nb-25Ti alloy.

Abstract Nowadays, as the oil reservoirs reaching their half-life, using enhanced oil recovery methods is more necessary and more common. Simulations are the synthetic process of real systems. In this study, simulation of water and surfactant injection into a porous media containing oil (two-phase) was performed using the computational fluid dynamics method on the image of a real micro-model. Also, the selected anionic surfactant is sodium dodecyl sulfate, which is more effective in sand reservoirs. The effect of using surfactant depends on its concentration. This dependence on concentration in using injection compounds is referred to as critical micelle concentration (CMC). In this study, an injection concentration (inlet boundary) of 1000 ppm was considered as a concentration less than the CMC point (2365 ppm). This range of surfactant concentrations after 4.5 ms increased the porous media recovery factor by 2.21%. Surfactant injection results showed the wettability alteration and IFT finally increases the recovery factor in comparison with water injection. Also, in wide channels, saturation front, and narrow channels, the concentration front has a great effect on the main flowing.

The influence of interfacial energy anisotropy on dendrite morphology is still not well understood. We investigated the dendrite morphology and solute segregation in Al-Zn alloys using the phase-field method. We observed typical and atypical dendrite orientation transition and three types of dendrite morphology “Christmas tree dendrite”, “seaweed dendrite” and “butterfly dendrite” depending on the anisotropic interfacial energy. Two types of dendrite may undergo the so called-“smooth dendrite orientation transition”, which produces a large amount of solute segregation holes in 〈100〉 dendrites, and liquid grooves in 〈110〉 seaweed and butterfly patterns, thus may significantly impact the mechanical properties of castings. We identified a strong correlation between supersaturation and the formation of liquid holes and liquid grooves as well as the degree of solute segregation. We also proposed that the interfacial energy anisotropy can be related with Zn content by discussing the anisotropic Young’s modulus and experimental pattern of Al-Zn alloys. The inducement of anisotropy and the quantitative relationship between interfacial energy anisotropy, Zn concentration, elastic anisotropy and dendrite orientation deserves attention.

The influence of periodic pressure with low and high frequencies on microstructure and dendritic sidebranching was studied by using 3-D phase field method. In both low and high frequency cases, the variation trend of SDAS (secondary dendritic arm spacing) with increasing pressure frequency is opposite to that of sidebranching frequency, while the variation trend of the average length of secondary arms is consistent with that of sidebranching frequency. The high sidebranching frequency indicates that more secondary arms share the whole driving force of dendrite growth, resulting in lower driving force for each one and leading to less developed secondary arms. The smallest SDAS is obtained when perturbed by the periodic pressure with the frequency of 0.157/τ0 (τ0 is the physical unit of time in the dimensionless phase field model) and 2.200/τ0 in low and high frequency cases, respectively. Comparisons of dendritic morphology and secondary arms are made between the low and high frequency cases. Firstly, in the low frequency case, secondary arms are luxuriant especially when pressure frequency is low, with many high-order side branches stretching out. Secondly, the average length of secondary arms in primary dendrite is longer in the low frequency case than that without pressure, and much longer than that in the high frequency case. Thirdly, the dendrite tip without side branches in the high frequency case is much longer than that in the low frequency case. All of the differences in dendritic morphology and sidebranching in the two cases can be attributed to the different modulation mechanism. In the low frequency case, periodic pressure determines tip velocity and then modulates sidebranching directly. While in the high frequency case, periodic pressure cannot determine sidebranching directly, but via modulating tiny protuberances in dendrite tip, part of which evolves into side branch. In this case, the tiny protuberances take part of the whole driving force, leading to less developed secondary arms.

The phase field method integrates the Griffith theory and damage mechanics approach to predict crack initiation, propagation, and branching within one framework. No crack tracking topology is needed, and complex crack shapes can be captures without user intervention. In this paper, a detailed description of how the phase field method is implemented with explicit dynamics into LS-DYNA is provided. The displacement field and the damage field are solved in a staggered approach and the phase field equation is solved every Nth time step (N is refered to as calculation cycle) to save computational time. An N value smaller than 1/400 of the total time step numbers is suggested. Several simulations are presented to demonstrate the feasibility of this solving scheme.

In this paper, the grain-refining effect of Al-Nb-B refiner in wrought Al alloys and corresponding refining mechanism was investigated using phase-field simulation and experiment methods. Through experimental statistics and data correction, the seed radius introduced by Al-Nb-B refiner into Al alloy melt and the corresponding quantitative density distribution data were obtained, and a seed density model (SDM) model has been established. On this basis, a multi-phase field method (MPFM) combined with the calculation of phase diagram (CALPHAD) has been employed to simulate the α -Al grain evolution of Al-Mg-Si alloy during the solidification process with the help of thermodynamic database. Experimental studies showed that the addition of 0.03 wt% Nb (Adding by Al-1.93Nb-0.22B master alloy) significantly reduced the grain size from 956.4 μ m to about 219.4 μ m, and the grain size slowly decreased to 192.3 μ m by continuing to add refiner to 0.09 wt% Nb. Meanwhile, the simulation results demonstrated that after the addition of refiners, dendrite morphology transformed from the relatively developed dendrites with secondary and tertiary dendrites to a fine and uniform equiaxed shape. Simulation and experimental studies showed quantitative agreement. In addition, the results also prove that the addition of Al-Nb-B refiner can provide high-quality and stable heterogeneous nucleation particles.

The tensile fracture is a widespread feature in rock excavation engineering, such as spalling around an opened tunnel. The phase field method (PFD) is a non-local theory to effectively simulate the quasi-brittle fracture of materials, especially for the propagation of a tensile crack. This work is dedicated to study the tensile failure characteristics of rock-like materials by the PFD simulation of the Brazilian test of the intact and fissure disk samples. The numerical results indicate that the tensile strength of the disk sample is anisotropic due to the influence of pre-existing cracks. The peak load decreases at first and then increases with the increase of the inclination angle, following the U-shaped trend. The simulation results also indicate that the wing crack growth is the main failure characteristic. Moreover, the crack propagation path initiates at the tip of the pre-existing crack when the inclination angle is less than 60°. Crack propagation initiates near the tip of the pre-existing crack when the angle is 75°, and it initiates at the middle of the pre-existing crack when the angle is 90°. Finally, all cracks extend to the loading position and approximately parallel to the loading direction. This process is in agreement with the Brazilian test of pre-existing cracks in the laboratory, which can validate the effectiveness of the PFD in simulating the tensile fracture of rock-like materials. This study can provide a reference for the fracture mechanism of the surrounding rock in the underground excavation.

In this paper, a thermomechanical coupled phase field method is developed to model cracks with frictional contact. Compared to discrete methods, the phase field method can represent arbitrary crack geometry without an explicit representation of the crack surface. The two distinguishable features of the proposed phase field method are: (1) for the mechanical phase, no specific algorithm is needed for imposing contact constraints on the fracture surfaces; (2) for the thermal phase, formulations are proposed for incorporating the phase field damage parameter so that different thermal conductance conditions are accommodated. While the stress is updated explicitly in the regularized interface regions under different contact conditions, the thermal conductivity is determined under different conductance conditions. In particular, we consider a pressure-dependent thermal conductance model (PDM) that is fully coupled with the mechanical phase, along with the other three thermal conductance models, i.e., the fully conductive model (FCM), the adiabatic model (ACM), and the uncoupled model (UCM). The potential of this formulation is showcased by several benchmark problems. We gain insights into the role of the temperature field affecting the mechanical field. Several 2D boundary value problems are addressed, demonstrating the model’s ability to capture cracking phenomena with the effect of the thermal field. We compare our results with the discrete methods as well as other phase field methods, and a very good agreement is achieved.

In comparison to Pb-based solders which have a toxic effect, the tin-silver-copper (SAC) family of alloys have relatively strong reliability and are widely used in the electronics industry. Phase separation and coarsening phenomenon on the surface of 96.5 wt. % Sn-3.0 wt. % Ag-0.5 wt. % Cu (SAC305) solder products exhibit special microstructural features and offer opportunities for the microstructure control of microelectronic interconnects. However, the formation mechanism of such morphological patterns is still unknown. Here, we applied a combination of experimental and phase field methods to study how such patterns form. It was observed that the pattern was Sn-rich and exhibited the characteristic morphology of spinodal decomposition. Contrary to earlier findings that only binary systems like Sn-Pb and Sn-Bi experienced such phenomena, spinodal decomposition was firstly observed in ternary solder system Sn-Ag-Cu. Morphology of Sn-rich patterns depended on whether the spinodal decomposition reacted completely. SAC305 solder alloy was easily decomposed by Sn component after being heated to roughly 260 °C. The above conclusions could offer theoretical support for quantitatively controlling the microstructure of solder alloys and would enhance the quality of related products.

This work simulates the morphological evolution process of the solidification interface of silicon crystal. Based on the phase field model of single dendrite growth of pure material in a single-phase system, the control equation of the phase field is re-optimized, and an interface free energy anisotropy equation that can simulate the competitive growth of multiple crystal grains is established. The competitive growth of polysilicon is then simulated and analyzed. The results show that when the degree of undercooling exceeds a certain value, the non-facet crystals are transformed into facet crystals. The main branches in each direction are relatively thick when the anisotropy is small. With increasing anisotropy, the main branches in each direction show gradual thinning, and edges and corners appear on the interface. The dendrites are no longer smooth, and they transform from non-facet crystals to facet crystals. The main branches of different grains inhibit each other when multiple crystal grains compete for growth. The growth of the main branches is curved, which is different from existing branches. The experimental results can more realistically simulate the evolution process of single-crystal silicon and polycrystalline silicon crystal micromorphology.

The spinodal decomposition was studied in the aged Mn-40 at. %Cu, Mn-30 at. %Cu, Mn-20 at. %Cu alloys using a phase-field model based on the Cahn–Hillard equation, considering a subregular solution model and the energy contribution of the magnetic behavior. The simulations were performed at aging temperatures of 300, 400, and 500 °C for times from 1 to 240 min. The growth kinetics of the Mn concentration profiles with time indicated clearly that the phase decomposition of the supersaturated solid solution γ into a mixture of Mn-rich γ′ and Cu-rich γ phases occurred by the spinodal decomposition mechanism. Moreover, the phase decomposition at the early stages of aging exhibited the characteristic morphology of spinodal decomposition, an interconnected and percolated microstructure of the decomposed phases. The most rapid growth kinetics of spinodal decomposition occurred for the aging of Mn-20 and 30 at. %Cu alloys because of the higher driving force. The presence of the phase decomposition is responsible for the increase in hardness, as well as the improvement of the damping capacity of Mn-Cu alloys.

A generalized multiscale (micro-macro) finite element (FE) model for SiC-fiber reinforced SiC-matrix ceramic (SiCf/SiC) nuclear fuel claddings is established. In the macro level, the solid mesh of braided preform, which can be tailored by machine settings (braid angle, yarn width, and so on), is generated based on the braiding process simulation using the dynamic FE-solver, hiring the contact constraints. The matrix mesh and the yarn mesh are integrated by the embedded region constraint, with which the meshing difficulties can be avoided. In the micro-UD model, the progressive damage of the ceramic matrix is modeled using the phase field method (PFM) and the fracture is captured by Mohr–Coulombs criterion, which are stable and efficient in the description of the brittle crack initiation, coalition, and branching. Based on this multiscale model, the mechanical behavior of the braided SiCf/SiC nuclear fuel cladding tube is studied in detail. The superiorities over the homogenized tube model are demonstrated, too.

Abstract The solidification of metals and alloys and the resulting microstructures, which as a function of thermal and solutal parameters can evolve as planar, cellular and dendritic, are important from a practical point of view, since they strongly influence the properties and quality of the final product. In many practical situations it is impracticable to develop analytical solutions permitting reliable predictions of microstructural growth during unsteady-state solidification conditions. The Phase Field method has become very popular and effective in modeling complex solid/liquid interfaces due to its ability to simulate the interface kinetics and the formation and evolution of different morphologies along the solidification process. In this work, a numerical analysis of the microstructural evolution during the transient solidification of dilute alloys of the Al-Cu-Si system is developed, which uses a phase-field approach for the simulation of ternary alloys. The phase-field, energy and solute concentration equations were numerically solved for the correspondent ternary system, varying the mesh parameters, temperature and alloy composition. The analysis performed were confronted with existing theoretical models and the results obtained are in agreement with the solidification theory.

The effect of shear flow on spherical nanoparticles (NPs) migration near a liquid–liquid interface is studied by numerical simulation. We have implemented a compact model through which we use the diffuse interface method for modeling the two fluids and the molecular dynamics method for the simulation of the motion of NPs. Two different cases regarding the state of the two fluids when introducing the NPs are investigated. First, we introduce the NPs randomly into the medium of the two immiscible liquids that are already separated, and the interface is formed between them. For this case, it is shown that before applying any shear flow, 30% of NPs are driven to the interface under the effect of the drag force resulting from the composition gradient between the two fluids at the interface. However, this percentage is increased to reach 66% under the effect of shear defined by a Péclet number Pe = 0.316. In this study, different shear rates are investigated in addition to different shearing times, and we show that both factors have a crucial effect regarding the migration of the NPs toward the interfacial region. In particular, a small shear rate applied for a long time will have approximately the same effect as a greater shear rate applied for a shorter time. In the second studied case, we introduce the NPs into the mixture of two fluids that are already mixed and before phase separation so that the NPs are introduced into the homogenous medium of the two fluids. For this case, we show that in the absence of shear, almost all NPs migrate to the interface during phase separation, whereas shearing has a negative result, mainly because it affects the phase separation.

The phase field model based on the nonlinear Cahn-Hilliard equation was applied to analyze the spinodal decomposition process in Al-Zn and Al-Zn-Cu alloys. Partial differential equations were solved using the explicit finite difference method for the Al- 20, and 35 at. % Zn alloys aged at temperatures between 25 and 100 °C for times from 10 s to 2000 s and Al-20at.%Zn-10at.%Cu and Al-20at.%Zn-5at.%Cu alloys at temperatures between 400 and 500 °C for times from 3600 to 360000 s. Thermo-Calc indicated that the copper addition extends the presence of the metastable miscibility gap up to a temperature of about 597 °C in comparison to the temperature of 350 °C for the binary case. This miscibility gap was calculated assuming that the equilibrium phases were not present and thus it is only existing at the early stages of aging. Simulation results pointed out that the phase decomposition process is much faster in the binary alloys than that in the ternary alloys in spite of the higher aging temperature for the latter case.

Bernaise (Binary Electrohydrodynamic Solver) is a flexible high-level finite element solver of two-phase electrohydrodynamic flow in complex geometries. Two-phase flow with electrolytes is relevant across a broad range of systems and scales, from “lab-on-a-chip” devices for medical diagnostics to enhanced oil recovery at the reservoir scale. For the strongly coupled multi-physics problem, we employ a recently developed thermodynamically consistent model which combines a generalized Nernst–Planck equation for ion transport, the Poisson equation for electrostatics, the Cahn–Hilliard equation for the phase field (describing the interface separating the phases), and the Navier–Stokes equations for fluid flow. We present an efficient linear, decoupled numerical scheme which sequentially solves the three sets of equations. The scheme is validated by comparison to cases where analytical solutions are available, benchmark cases, and by the method of manufactured solution. The solver operates on unstructured meshes and is therefore well suited to handle arbitrarily shaped domains and problem set-ups where, e.g., very different resolutions are required in different parts of the domain. Bernaise is implemented in Python via the FEniCS framework, which effectively utilizes MPI and domain decomposition. Further, new solvers and problem set-ups can be specified and added with ease to the Bernaise framework by experienced Python users.

This article studies the dielectric and piezoelectric behavior of unpoled and poled barium titanate (BaTiO3) polycrystals with oxygen vacancies. A phase field model is employed for BaTiO3 polycrystals, coupled with the time-dependent Ginzburg–Landau theory and the oxygen vacancies diffusion, to demonstrate the interaction between oxygen vacancies and domain evolutions. To generate grain structures, the phase field model for grain growth is also used. The hysteresis loop and butterfly curve are predicted at room and high temperatures. The permittivity, and longitudinal and transverse piezoelectric constants of the BaTiO3 polycrystals are then examined for various grain sizes and oxygen vacancy densities.

The superior fracture toughness of zirconia is closely correlated with stress-induced martensitic phase transformation around a crack tip. In this study, a modified phase field (PF) model coupling phase transformation and fracture is proposed to study the fracture behavior and toughening effect of tetragonal zirconia polycrystal (TZP). The stress-induced tetragonal to monoclinic (t–m) phase transformation around a static or propagating crack is characterized with PF simulations. It is shown that the finite size and shape of the transformation zone under different loads and ambient temperatures can be well predicted with the proposed PF model. The phase transformation may decrease the stress level around the crack tip, which implies the toughening effect. After that, crack propagation in TZP is studied. As the stress field is perturbed by the phase transformation patterns, the crack may experience deflection and branching in the propagation process. It is found that the toughness of the grain boundaries (GBs) has important influences on the crack propagation mode. For TZP with strong GBs, the crack is more likely to propagate transgranularly while, for TZP with weak GBs, intergranular crack propagation is prevalent. Besides that, the crystal orientation and the external load can also influence the topology of crack propagation.

The boundary heat flow has important significance for the microstructures of directional solidified binary alloy. Interface evolution of the directional solidified microstructure with different boundary heat flow was discussed. In this study, only one interface was allowed to have heat flow, and Neumann boundary conditions were imposed at the other three interfaces. From the calculated results, it was found that different boundary heat flows will result in different microstructures. When the boundary heat flow equals to 20 W·cm-2, the growth of longitudinal side branches is accelerated and the growth of transverse side branches is restrained, and meanwhile, there is dendritic remelting in the calculation domain. When the boundary heat flow equals to 40 W·cm-2, the growths of the transverse and longitudinal side branches compete with each other, and when the boundary heat flow equals to 100-200 W·cm-2, the growth of transverse side branches dominates absolutely. The temperature field of dendritic growth was analyzed and the relation between boundary heat flow and temperature field was also investigated.