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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.

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 growth of lithium dendrite in solid electrolytes has become a major obstacle to the development of solid-state lithium batteries. Lithium dendrite can cause problems such as reduced Coulombic efficiency and shortened lifespan of the battery, and may even cause short circuits that lead to battery failure. The phase field method is used to establish a coupled electro-thermo-mechanical model to study the growth of lithium dendrite, and the plastic behavior of lithium dendrite is also considered. Based on the theory of heat transfer models, the influence of temperature changes on the morphology of lithium dendrite and von Mises stress are analyzed. Using the theory of plastic work, the influence of temperature changes and external pressure changes on the plastic strain of lithium dendrite are analyzed. The results show that the inhibition effect on lithium dendrite is more significant with increasing external pressure and temperature values, and von Mises stress also increases. Lithium dendrite may fracture due to excessive von Mises stress and form dead lithium. The equivalent plastic strain of lithium dendrite increases with the increase of temperature and external pressure values. [ABSTRACT FROM AUTHOR]

A phase field method based on the Cahn‐Hilliard equation was used to establish a numerical model for the motion and coalescence of multiple droplets under an electric field. The influencing factors of the droplet coalescence rate, velocity field and pressure field distributions, and droplet cluster motion and coalescence were investigated. The observations showed that the dynamic behaviors of droplets under the action of the electric field can be characterized as not coming together, coalescence, and breakup after coalescence. High‐speed (low‐speed) and high‐pressure (low‐pressure) regions are generated inside the droplet or near the interface under the action of the electric field, and the size, number, and location of these regions change over time. The results of this study offer an essential guide for developing new electrostatic coalescers. [ABSTRACT FROM AUTHOR]

The phase field method (PFM) has been proposed and incorporated into the finite element method (FEM) for complex crack evolution problems. However, explicit cracks cannot be obtained in the phase field FEM (PFFEM). In the field of rock engineering, explicit cracks are indispensable for hydraulic fracturing problem in which crack opening displacement should be known, and compression-shear crack problems in which contact region should be determined. In this paper, the recently proposed phase field numerical manifold method (PFNMM) is developed to model rock fracturing processes. In PFNMM, PFM is regarded as a fracturing criterion, which deals with crack initiation, propagation, bifurcation and coalescence in a unified form; then crack paths are reconstructed and reproduced from smearing cracks; finally, the physical patches and manifold elements are cut with the reconstructed paths to obtain explicit cracks. The numerical results for several typical examples indicate that rock fracturing processes, including crack initiation without any preset cracks, crack propagation and crack merging, can be explicitly predicted. Besides, the results are in good agreement with the literature. Compared with PFFEM, explicit cracks and jump displacement fields across rock crack faces can be easily obtained by PFNMM. Highlights: The combination of PFM and NMM can reproduce the explicit fracture process in rock. The proposed method is capable of simulating the free opening and closing of cracks. The PFNMM can predict the evolution mechanism of multi-cracks and merging crack. [ABSTRACT FROM AUTHOR]

Natural crack structures significantly affect coal damage, and are a critical problem in mining engineering. In this paper, we introduced the fluorescent epoxy impregnation method (FEIM) to coal failure research, which can obtain cracks with an accuracy of 1 μm by a much lower experimental cost compared with CT scanning. Then we proposed a way to combine FEIM method and the phase-field finite element method (FEM), and established the tensor characterization theory of crack propagation to fully explore the effects of fracture structures on coal failure properties. The results demonstrated that the distribution of initial cracks in coal significantly affected the subsequent failure path at the micro level. Initiation of cracks occurred at the tips of inflection points of existing cracks. The cracks mainly propagated along or perpendicular to existing cracks. The cracks were tightly connected, forming larger crack networks and developing new cracks. The crack tensor theory can describe the crack initiation and propagation process well. The trace of the crack fabric tensor is equal to the total length of all cracks, while the crack orientation tensor describes the direction characteristics of the crack field. Natural cracks in coal greatly affected the initiation location, but the ultimate failure pattern was always shear failure. The proposed method combined experiments and numerical simulations can provide a basic reference for mining engineering. [ABSTRACT FROM AUTHOR]

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.

In this paper, a new method which combines the back-propagation artificial neural network and phase field method is proposed to predict the residual useful life of mechanical structures. Firstly, the phase field model of brittle fracture is constructed by phase-field method, and the data set about the corresponding strain value and lifespan of the mechanical structure is obtained. Then, the data set is employed as the training samples by back-propagation artificial neural network to obtain the lifespan prediction mathematical model of mechanical structure. With the help of prediction model, only the corresponding structure strain value can be used to predict the remaining life of the structure. Finally, the prediction performance of the proposed method is evaluated by numerical examples. The results show that the proposed method combining the phase-field method and neural network technology for predicting the remaining life of the structure has high accuracy. [ABSTRACT FROM AUTHOR]

A novel explicit phase field material point method (ex‐PFMPM) is proposed for modeling dynamic fracture problems. The rate‐dependent phase field governing equation is discretized by a set of particles, and the phase field is updated by the explicit forward‐difference time integration. Furthermore, the stability of the ex‐PFMPM is studied. A novel explicit critical time step formula is obtained based on the system eigenvalues in one dimension and then extended to two and three dimensions. The critical time step formula takes the effect of particle position and neighboring cell interaction into consideration, and can also be used in an explicit phase field finite element method. Several numerical examples, including a dynamic crack branching, a plate with pre‐existing crack under velocity boundary conditions and a three point bending problem are studied to verify the proposed ex‐PFMPM. The use of the history field in the explicit method is studied, which shows that it will lead to fake phase field update and overestimation of the fracture energy in the unloading case. All of the numerical results show that the proposed ex‐PFMPM has the capacity of modeling the crack initiation and propagation problems with both accuracy and efficiency. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We discuss the interaction of the strongly nonlinear fluid motion induced by the collapse of a vapor microbubble over a planar surface and the elastic dynamics of the underlying solid. The fluid is described using an extension of the Navier-Stokes equations endowed with distributed capillary stresses in the context of a diffuse interface approach. The collapse of the bubble is triggered by overpressure in the liquid and leads to an intense jet that pierces the bubble, changing the bubble topology from spheroidal to toroidal, and impinges the solid wall inducing an intense and strongly localized load. Moreover, at bubble collapse, a compression wave is launched into the liquid surrounding the bubble. By propagating along the solid surface, the compression wave combined with the liquid jet excites the dynamics of the elastic solid, producing a complex system of waves, including, longitudinal, transversal, and Rayleigh waves, propagating in the solid. It is conjectured that the intense deformation of the solid induced by the strongly localized liquid jet may lead to the plastic deformation of the solid producing the surface pitting observed in many applications subject to cavitation-induced material damage. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The dynamic growth process of intermetallic compounds (IMCs) at the interface of copper–tin coating and the process of microvoid formation were studied using the phase field method combined with thermal–mechanical–electrical diffusion coupling theory. Numerical results show that the thickness of IMCs in the copper–tin coating increased rapidly and then slowly. Current density had a significant effect on the growth of IMCs at the interface. Under the action of electromigration, the thickness of IMCs at the anode was much greater than that at the cathode, which was consistent with the experimental results. The thickness of IMCs increased with the increase in current density. The influence of a rough interface on IMCs was also studied. Uneven diffusion occurred in the rough interface, resulting in microvoids, and the electromigration effect was one of the reasons for the unbalanced diffusion. [ABSTRACT FROM AUTHOR]

Copyright of Acta Materiae Compositae Sinica is the property of Acta Materiea Compositae Sinica Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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.

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 FROM AUTHOR]

Copyright of Natural Gas Industry is the property of Natural Gas Industry Journal Agency and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A critical issue in simulating the evolution of a crack using the phase field approach resides in the ability of the model to predict crack nucleation. In this work, we study the impact of the Ambrosio–Tortorelli regularized crack formulation on the phase field model's ability to represent the phenomena of nucleation and early crack propagation in two-dimensional domains under generic boundary conditions. Through numerical experiments we study the nature of the errors introduced by the crack's regularization at the early stages of crack propagation and its critical effects on the predictability of the nucleation phenomenon. We also show that the errors introduced through the regularization of the crack are systematic and can be predicted for a broad range of boundary conditions. Both external and internal crack cases are analyzed and the scalability of the error is verified. We then investigate the nature of the error and identify its localization with reference to the crack geometry. We conclude that the classical formulation of phase field is not designed to predict nucleation without affecting the early propagation of the crack. [ABSTRACT FROM AUTHOR]

Interphase mass transfer is an important solute transport process in two-phase flow in porous media. During two-phase flow, hydrodynamically stagnant and flowing zones are formed, with the stagnant ones being adjacent to the interfaces through which the interphase mass transfer happens. Due to the existence of these stagnant zones in the vicinity of the interface, the mass transfer coefficient decreases to a certain extent. There seems to be a phenomenological correlation between the mass transfer coefficient and the extent of the stagnant zone which, however, is not yet fully understood. In this study, the phase-field method-based continuous species transfer model is applied to simulate the interphase mass transfer of a dissolved species from the immobile, residual, non-aqueous phase liquid (NAPL) to the flowing aqueous phase. Both scenarios, this of a simple cavity and this of a porous medium, are investigated. The effects of flow rates on the mass transfer coefficient are significantly reduced when the stagnant zone and the diffusion length are larger. It is found that the stagnant zone saturation can be a proxy of the overall diffusion length of the terminal menisci in the porous medium system. The early-stage mass transfer coefficient continuously decreases due to the depletion of the solute in the small NAPL clusters that are in direct contact with the flowing water. The long-term mass transfer mainly happens on the interfaces associated with large NAPL clusters with larger diffusion lengths, and the mass transfer coefficient is mainly determined by the stagnant zone saturation. Article highlights: Flow rates can hardly affect the mass transfer coefficient when the stagnant zone and the diffusion length are large. The stagnant zone saturation can be a proxy of the overall diffusion length of the menisci in the porous media system. The correlations between the mass transfer coefficient and the stagnant zone saturations are built, based on simulations at different flow rates. [ABSTRACT FROM AUTHOR]

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.

Neurodevelopmental disorders (NDDs) have arisen as one of the most prevailing chronic diseases within the US. Often associated with severe adverse impacts on the formation of vital central and peripheral nervous systems during the neurodevelopmental process, NDDs are comprised of a broad spectrum of disorders, such as autism spectrum disorder, attention deficit hyperactivity disorder, and epilepsy, characterized by progressive and pervasive detriments to cognitive, speech, memory, motor, and other neurological functions in patients. However, the heterogeneous nature of NDDs poses a significant roadblock to identifying the exact pathogenesis, impeding accurate diagnosis and the development of targeted treatment planning. A computational NDDs model holds immense potential in enhancing our understanding of the multifaceted factors involved and could assist in identifying the root causes to expedite treatment development. To tackle this challenge, we introduce optimal neurotrophin concentration to the driving force and degradation of neurotrophin to the synaptogenesis process of a 2D phase field neuron growth model using isogeometric analysis to simulate neurite retraction and atrophy. The optimal neurotrophin concentration effectively captures the inverse relationship between neurotrophin levels and neuron survival, while its degradation regulates concentration levels. Leveraging dynamic domain expansion, the model efficiently expands the domain based on outgrowth patterns to minimize degrees of freedom. Based on truncated T-splines, our model simulates the evolving process of complex neurite structures by applying local refinement adaptively to the cell/neurite boundary. Furthermore, a thorough parameter investigation is conducted with detailed comparisons against neuron cell cultures in experiments, enhancing our fundamental understanding of the possible mechanisms underlying NDDs. [ABSTRACT FROM AUTHOR]

• Develop a phase field method for assessing the threshold conditions to initiate hydrogen-induced cracks on dented pipelines. • Define the influences of dent depth, initial hydrogen concentration and internal pressure on crack initiation. • Determine the threshold values of dent depth and internal pressure to initiate cracks under hydrogen impact. An accurate determination of the threshold conditions to initiate cracks on aged hydrogen pipelines is paramount for ensuring energy transport safety. In this work, a finite element-based phase field method was developed to assess the crack initiation on dented pipelines while considering the hydrogen (H) impact. Theoretical and multi-physics numerical formulas were derived for prediction of the elastic-plastic fracture behavior of H-contained steel. A critical phase field parameter, ϕ =0.69, is defined for predicting crack initiation at the dent on pipelines. The presence of H within the steel decreases the threshold dent depth for initiating H-induced cracks. When the initial H concentration increases from 0 to 0.5 wppm, the maximum dent depth for crack initiation reduces from 17.5 mm to 10.7 mm. The maximum dent depth required for crack initiation reduces from 17.5 mm to 7.8 mm when an internal pressure of 8 MPa is applied on the steel pipe. The site with the maximum phase field parameter changes during indentation, implying that the location initiating cracks depends on the dent dimension. The existing criteria in ASME B31.12 standard are not applicable for predicting H-induced crack initiation on dented pipelines. This study proposes a new method to predict hydrogen-induced cracking on aged pipelines when transporting hydrogen. [Display omitted] [ABSTRACT FROM AUTHOR]

• A novel acceleration method for simulating multiphase flows based on spectral element method and phase field method. • This method reduces the dimensionality of linear algebraic equation system, resulting in faster computations. • The acceleration method can improve computational efficiency by at least 49.5% for solving the phase field equation and at least 70.2% for computing two-phase flow, while maintaining accuracy. • The accuracy and efficiency of the present method for two-phase flows at high Re and large density/viscosity contrast are verified. In this manuscript, we unveil an innovative acceleration technique for the simulation of multiphase flows, which builds upon the foundation laid by our previously devised multiphase flow solver. The cornerstone of this method lies in augmenting computational efficiency through the selective updating of variables solely within domains characterized by significant gradients of the phase variable. This tactic diminishes the dimensionality of the system of linear equations, thereby hastening the computational process. To pinpoint the regions warranting focused attention, a judicious criterion is indispensable to strike an optimal balance between efficiency and precision. This criterion affords a marked decrement in computational expenditure while preserving the fidelity of the original methodology. Rigorous validations corroborate that this acceleration mechanism can enhance computational efficiency by a minimum of 49.5% in resolving the phase field equation and by at least 70.2% in the computation of pragmatic two-phase flows, without compromising accuracy. Moreover, the efficacy of this acceleration technique is inversely proportional to the rate of interface evolution, becoming increasingly efficient as the interface evolves more slowly. [Display omitted] [ABSTRACT FROM AUTHOR]

Tensile crack-induced collapse is widely distributed in major mountainous areas of China, often result in sudden and catastrophic consequences due to the rapid movement of the falling bodies. To better understand the development of tensile crack-induced collapses, the phase field method, originally used for studying brittle fracturing, has been further applied in this research. Firstly, the application of the phase field method in modeling rock tensile failure is validated, demonstrating its effectiveness. The rationality of applying the phase field method for analyzing tensile crack-induced collapses has been thoroughly investigated, and solutions to address phase field disorder have been proposed. By incorporating heterogeneity features and implementing a prefabricated crack, both approaches can effectively address the issue of phase field disorder. However, the latter method aligns more closely with engineering practices. The present study provides a comprehensive investigation into the development of tensile crack-induced collapses with varying cavity morphologies. The corresponding change laws of tensile crack-induced collapses are systematically analyzed and summarized. Moreover, novel stability evaluation parameters, including reduction coefficients and safety points, are proposed and successfully applied in calculations. Our research on different cave morphologies has shown that the depth and shape of a cave, as well as the thickness and center of gravity of protruding rock masses, significantly influence the development process of tensile crack-induced collapse. This approach facilitates a comprehensive examination of the fracture process and yields valuable insights into the mechanics underlying tensile crack-induced collapses. [ABSTRACT FROM AUTHOR]

• Fracture toughness and modes of C/SiC have temperature and orientations dependence. • The work of fracture is the largest with a unique fracture mode at 1600 ℃. • The mechanism of crack propagation is different with the increase of temperature. • A phase field model for high-temperature fracture simulation of C/SiC is proposed. The high-temperature fracture toughness of C/SiC composites is of great significance for the tolerance assessment and safety application of components in service. In this work, combining the high-temperature experimental technique and phase field method, the fracture toughness of PIP-C/SiC composites at 25–1600 ℃ in inert atmosphere was tested, and the microcrack propagation at different temperatures was simulated. The fracture model considers the effects of residual stress, temperature-dependent properties of constituents and interfacial graphitization on the crack propagation process. The results show that the fracture toughness and modes of C/SiC composites have significant temperature dependence and difference in in-plane and out-of-plane orientations. As the increase of temperature, the cracks exhibit three typical modes of deflection, penetration and long-debonding at the fiber–matrix interphase. It is worth noting that C/SiC composites show a unique fracture mode at 1600 ℃ with the work of fracture increasing significantly. Overall, the work provides a guidance for the damage tolerance assessment of C/SiC composites in engineering. [ABSTRACT FROM AUTHOR]

Energetic Variational Approach is used to derive a novel thermodynamically consistent three-phase model of a mixture of Newtonian and visco-elastic fluids. The model which automatically satisfies the energy dissipation law and is Galilean invariant, consists of coupled Navier-Stokes and Cahn-Hilliard equations. Modified General Navier Boundary Condition with fluid elasticity taken into account is also introduced for using the model to study moving contact line problems. Energy stable numerical scheme is developed to solve system of model equations efficiently. Convergence of the numerical scheme is verified by simulating a droplet sliding on an inclined plane under gravity. The model can be applied for studying various biological or biophysical problems. Predictive abilities of the model are demonstrated by simulating deformation of venous blood clots with different visco-elastic properties and experimentally observed internal structures under different biologically relevant shear blood flow conditions.

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.

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.

The phase field method uses a length scale parameter to regularize the discrete crack to a diffuse crack, which removes the numerical tracking of the discontinuities in the displacement. The displacement field is coupled with the phase field and both are solved as a sequentially coupled systems using staggered method. The phase field ϕ varies between zero and unity (i.e., ϕ = 0 for intact region and ϕ = 1 for fully broken region), and it is a scalar. In this study, a new way of implementation is done using ABAQUS software to solve for the two fields. User defined element subroutine (UEL) is used to solve for the phase field variable and user defined material subroutine (UMAT) for the displacement field variable. Phase field model can simulate any complex crack paths and branching even without previously defined cracks. Some benchmark examples of quasi-static brittle fracture and dynamic brittle fracture are solved and verified with the existing numerical results. To account for the rate-dependent effect under high-rate loading, micro-inertia is incorporated into the phase-field model for dynamic fracture as proposed in the literature and verified with one example. [ABSTRACT FROM AUTHOR]

Fracture initiation, propagation, merging and branching in rocks are considerably complicated and commonly exhibit a great variety of patterns. How to characterize and capture the complicated fracturing process is still a challenging task for recent studies. In this study, a new double phase field model (DPFM) for mixed-mode fractures in rocks modified from triple shear energy criterion is proposed. We introduce two independent phase field variables to describe the tensile fractures and shear fractures individually. In the proposed DPFM, a new fracture driving force based on triple shear energy criterion is proposed and a hybrid formulation is adopted for constructing the new fracture governing equations. Differently from the existing phase field models, our new model can comprehensively consider the influence of cohesion, internal friction angles and intermediate principal stress. Uniaxial compression tests on rock specimens with an inclined flaw and double flaws are performed to validate our DPFM. We also test the fracture patterns of rocks with intermittent fissures in 3D compressive-shear tests. It is observed that the proposed new DPFM can capture the complicated fracturing processes and fracture coalescence types, which are in good agreement with the existing experimental and numerical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Unlike the lattice Boltzmann method (LBM), the lattice Boltzmann flux solver (LBFS) is free from the limitation of uniform meshes, coupled time step and mesh spacing, and high memory cost. Specifically, the existing interface LBFS (LBFS-CH0) is reconstructed by locally applying the Cahn–Hilliard-based (CH-based) LBM with additional artificial terms. In the framework of LBM, the influence of the additional artificial terms has been assessed. Besides, the comparative study of the CH-based and the Allen–Cahn-based (AC-based) LBM shows that the latter has higher accuracy and stability. Though the LBFS is originated from the LBM, their performances may not be exactly the same. In this study, the AC-based and CH-based LBFS (LBFS-AC and LBFS-CH1) models which can accurately recover the phase field equations are developed from the interface LBM models without any additional terms. The gradient term in the LBFS-AC is discretized by the second-order central difference method. Compared with the LBFS-CH0, the LBFS-CH1 can eliminate the additional terms and recover the true CH equation. Then these interface LBFS models are compared to evaluate their performance and the influence of the additional terms. Different from the LBM, it is found that the CH-based LBFS is superior to the AC-based one in terms of the numerical accuracy and stability. In addition, numerical results show that the additional terms have minor effect on the ability to model the phase interface. This study can provide guidance to ensure the accuracy and stability of interface modeling. Taking into account the accuracy, stability and simplicity, we can conclude that the LBFS-CH0 model could be the first choice to simulate the phase interface in the framework of LBFS. [ABSTRACT FROM AUTHOR]

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.

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.

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.

The behavior of crack initiation, propagation and coalescence in rock is the key mechanics of rock deformation, strength and stability, and it is important to predict and analyze crack growth behavior by numerical method. Phase field（PF） method overcomes the challenge of the traditional fracture mechanics theory and numerical method to explicitly track the crack surface within the framework of strict and accurate theory, and can simulate crack initiation, propagation, branching and multi-cracking. A multi-scale model that can reflect the internal microstructure of rock materials was established at the main area of crack growth. The failure progress of crack initiation, propagation and coalescence within the basalt under three-point bending test was predicted and analyzed by phase field method. The influence of the distribution of mineral inclusions and energy release rate on the crack propagation behavior was investigated and the numerical simulation results were compared and verified with the experimental data. It is shown that the proposed multi-scale PF model considering the complex internal microstructure of basalt rock is able to simulate and predict the progress of the crack behavior. The behavior of crack initiation, propagation and coalescence in rock is the key mechanics of rock deformation, strength and stability, and it is important to predict and analyze crack growth behavior by numerical method. Phase field（PF） method overcomes the challenge of the traditional fracture mechanics theory and numerical method to explicitly track the crack surface within the framework of strict and accurate theory, and can simulate crack initiation, propagation, branching and multi-cracking. A multi-scale model that can reflect the internal microstructure of rock materials was established at the main area of crack growth. The failure progress of crack initiation, propagation and coalescence within the basalt under three-point bending test was predicted and analyzed by phase field method. The influence of the distribution of mineral inclusions and energy release rate on the crack propagation behavior was investigated and the numerical simulation results were compared and verified with the experimental data. It is shown that the proposed multi-scale PF model considering the complex internal microstructure of basalt rock is able to simulate and predict the progress of the crack behavior. [ABSTRACT FROM AUTHOR]

In this paper, melting of long Al nanowires is studied using a phase field model in which deviatoric transformation strain described by a kinetic equation produces a promoting driving force for both melting and solidification and consequently, a lower melting temperature is resolved. The coupled system of the Ginzburgâ€"Landau equation for solidification/melting transformation, the kinetic equation for the deviatoric transformation strain and elasticity equations are solved using the COMSOL finite element code to obtain the evolution of melt solution. A deviatoric strain kinetic coefficient is used which results in the same pressure as that calculated with the Laplace equation in a solid neglecting elastic stresses. The surface and bulk melting temperatures are calculated for different nanowire diameters without mechanical loading which shows a good agreement with existing MD and analytical results. For radii R > 5 nm, a complete surface solid-melt interface is created which propagates to the center. For smaller radii, premelting occurs everywhere starting from the surface and the nanowire melts without creating the interface. The melting rate shows an inverse power relationship with radius for R < 15 nm. For melting under pressure, the model with constant bulk modulus results in an unphysical parabolic variation versus pressure in contrast to the almost linear increase of the melting temperature versus pressure from known MD simulations. Such drawback is resolved by considering the pressure dependence of the bulk modulus through the Murnaghan’s equation due to which an almost linear increase of the melting temperature versus pressure is obtained. Also, a reduction of the interface width and a significant increase of the melting rate versus pressure are found. The presented model and results allow for a better understanding of the premelting and melting of different metallic nanowires with various loading conditions and structural defects. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, a novel phase-field (PF) model is proposed to study the thermal aging mechanism of single crystalline t'-YSZ. The influences of the initial compositional content of yttria and the initial twin structure of the t' phase on the aging process are systematically discussed. The PF model can recover the modulated structure and nano/micro hybrid structure observed in experiments. The PF simulation results indicate that the initial compositional content of yttria is the most important influential factor of the thermal aging process. Besides that, the transformation strain, the initial twin structure and the anti-phase boundaries (APBs) of the t' phase can also have significant influences on the thermal aging kinetics. The typical spinodal region is more suitable to predict the thermal aging behavior of single domain YSZ. For multi-domain YSZ with initial twin structures and APBs, the spinodal region should be further divided into the kernel region and marginal region. In the kernel region, the thermal aging occurs by spinodal decomposition with the formation of a modulated structure, which is followed by merging and coarsening. In the marginal region and outside the spinodal region, the phase decomposition leads to a hybrid structure with coarse grained cubic phase and fine grained tetragonal phase, which exhibits the characteristics of nucleation and growth. The hybrid structure is consistent with previous experimental observations. It is revealed that the boundaries of the nano sized tetragonal grains evolve from the twin boundaries and APBs. The nucleation-growth mechanism should be properly understood when it is applied to illustrate the evolution process of the hybrid structure. The PF model and the new insights obtained in this study are helpful to understand the thermal aging mechanisms of t'-YSZ. [ABSTRACT FROM AUTHOR]

As an environmentally friendly and efficient separation technology, electrostatic dehydration is widely used in oil-water separation in water-in-oil emulsions. To investigate droplet coalescence behavior under different electric field parameters, the coalescence of two water droplets in water-in-oil emulsions was numerically analyzed with the proprietary software Comsol Multiphysics. The motion of the interface was captured by the phase field method and microscopic mechanism of electrostatic coalescence in water-in-oil emulsions was determined. According to force simulation, the difference between electrostatic attraction and interfacial tension of two droplets' proximal-end and distal-end promoted their approach and fusion process. Furthermore, approach time, fusion time, coalescence time, and distal-end distance S were defined to describe the coalescence characteristics under different electric field frequencies, intensities, and waveforms. The drop-drop coalescence effect was more obvious under an electric field frequency of 20–40 Hz, a higher electric field intensity, as well as DC pulse and half-sinusoidal wave. The deformation degree and electric field intensity at the beginning of droplet coalescence had a significant influence on the coalescence velocity. The results obtained provide much knowledge and a theoretical basis for the technology of compact electrostatic coalescers, which is of great guiding significance for performance prediction and structure optimization. [ABSTRACT FROM AUTHOR]

Although the laser cladding technology is widely used in various fields, the laser cladding layer is prone to pores, inclusions and microcracks, severely limiting the popularization and application of this technology. High-frequency mechanical vibration can effectively reduce the micro defects in the cladding layer and strengthen the solidification structure of the cladding layer. In this paper, a laser cladding system for a disc laser assisted by ultrasonic vibration is built, and the dynamic response of ultrasonic vibration of the cladding experiment platform is tested. A finite element model of the dynamic response of the cladding experiment platform under ultrasonic vibration was established, and the difference of the vibration of the experimental platform substrate at different positions under the ultrasonic vibration was calculated. The internal amplitude of the substrate is differentiated in a hemispherical shape with the vibration source as the center and is distributed under the influence of ultrasonic vibration. On this basis, the phase field method was used to establish the ultrasonic-assisted disc laser cladding solidification model to obtain the dendrite growth state during the solidification process. The calculation reveals the change law of single crystal and polycrystalline solidification growth under the influence of ultrasonic amplitude at different substrate positions. Calculations show that ultrasonic vibration can effectively increase the growth rate of dendrites and promote the formation of secondary dendrite arms. The polycrystalline solidification growth model is closer to reality. The competitive growth of dendrites suppresses the growth of primary dendrites to a certain extent and affects the selective growth of dendrites. The results show that the larger the ultrasonic amplitude, the more rapid the formation of secondary dendrite arms. [ABSTRACT FROM AUTHOR]