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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. [ABSTRACT FROM AUTHOR]

Recent years, the polarization response of ferroelectrics has been entirely studied. However, it is found that the polarization may disappear gradually with the continually applied of electric field. In this paper, taking K0.48Na0.52NbO3(KNN) as an example, it was demonstrated that the residual polarization began to decrease when the electric field frequency increased to a certain extent using a phase-field methods. The results showed that the content of out-of-plane domains increased first and then decreased with the increase of applied electric field frequency, the maximum polarization disappeared at high frequencies, and the hysteresis loop became elliptical. In order to further study the abnormal changes of hysteresis loops of ferroelectrics under high electric field frequency, we analyzed the hysteresis loop and dielectric response of solid solution 0.1SrTiO3-0.9K0.48Na0.52NbO3. It was found that the doped hysteresis loop maintained its shape at higher frequency and the dielectric constant increased. This kind of doping has a higher field frequency adaptability, which has a key guiding role in improving the dielectric properties of ferroelectric thin films and expanding the frequency application range of ferroelectric nano memory. [ABSTRACT FROM AUTHOR]

The thermal–mechanical–electrical-diffusion strongly coupled equations are established, and the phase field method is employed to investigate the dynamic simulation of intermetallic compound (IMC) evolution at the Cu/Sn/Cu micro-solder joint interface. By comparing with experimental results on IMC evolution in micro-solder joints, this study explores the evolution law of IMCs under the influence of current, temperature, and applied load coupling. Both simulation and experimental findings indicate that higher current density promotes IMC formation, with the largest average thickness observed at a current density of 5000 A/m2. When the analog current density is 5000 A/m2, the anode IMC has a rapid growth period at 62 h, and the corresponding cathode has a rapid consumption period; when the analog current density is 4000 A/m2, the anode has the same phenomenon in 95 h; when the analog current density is 3000 A/m2, this phenomenon is not evident. The observed phenomena are consistent with the evolutionary trend of IMC from the fourth to the fifth day of the experiment, considering it may be affected by 'electronic wind stress' which makes the cathode IMC migrate to the anode. The average thickness of IMCs with an analog ambient temperature of 323–353 K is the largest, which is most suitable for the growth of IMCs. The experimental results also indicate that IMC growth is the fastest at 353 K. The anode IMC was inhibited by tensile stress and shear stress, but the effect on cathode was not obvious. [ABSTRACT FROM AUTHOR]

The tensile failure of complex branching cracks will cause irreversible damage to the structures. In this paper, the effects of the crack orientations, the local branch lengths, and the different load forms on the crack propagation pattern and the bearing capacity of structures are investigated based on the phase field method and experimental research method. The results show that the bearing capacity and propagation patterns of the specimens change significantly with the change of the orientation of the Y-shaped crack under tensile load. The local branch length of the crack only affects the sequence in which the regional crack reaches the final failure and the bearing capacity of the specimen. Under the dynamic impact load, the bifurcation phenomenon occurs, and with the increase of the dynamic impact load, the number of bifurcations increases. The crack propagation pattern in the numerical simulation is completely consistent with the experiment, and the difference in the maximum reaction force is 2.5%. [ABSTRACT FROM AUTHOR]

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 paper presents an algorithm for three-dimensional modelling of two-phase flows on the scale of pore size order for numerical evaluation of relative phase permeability curves of porous materials. Such an evaluation is performed based on the results of numerical simulation of primary drainage with subsequent waterflooding. In this case, models of porous materials based on three-dimensional tomographic images of rocks are used. The simulation of the flow considers the Stokes equation and the Cahn–Hilliard equation for modelling phase transfer, which allows us to determine phases using the concentration function. The combination of the phase field method and finite difference method makes it possible to correctly take into account the contact angle and stably calculate surface tension forces in domains with complex topology. [ABSTRACT FROM AUTHOR]

Water freezing in rock fractures causes volumetric expansion and fracture development through frost heaving. This study introduces a novel analytical model to investigate how uneven freezing force and surrounding rock pressure influence fracture initiation, based on mass conservation, elasticity, and water-ice phase transition principles. A model for rock fracture initiation considering freezing temperature, uneven freezing expansion, in-situ stress, and lateral pressure was proposed based on fracture mechanics. Equations for stress intensity factors were developed and validated using the phase field method. The effects of rock elastic modulus anisotropy and critical fracture energy density on fracture initiation were also discussed. The results show that the values of KI and KII exhibit an upward trend as the freezing temperature, uneven expansion, in-situ stress, and lateral pressure increase. The uneven freezing expansion has the most significant influence on KI and KII values among these parameters. As the uneven freezing expansion coefficient increases to 0.5, the fracture initiation mode shifts from tensile fracture to shear fracture. As the lateral pressure coefficient increases to 1, the fracture initiation mode shifts from tensile fracture to shear fracture. Rock elastic modulus anisotropy causes fractures to propagate in a clockwise direction, forming a 'butterfly' pattern. Critical fracture energy density an isotropy causes counterclockwise deviation in propagation direction, resulting in branching paths and an 'H'-shaped pattern. [ABSTRACT FROM AUTHOR]

In response to the problem that the standard AT2 phase field model cannot effectively model quasi-brittle failure and the existence of length dependence, a new modified phase field model is presented in this paper. By introducing an additional energy, the competing relationship between elastic strain energy and dissipation energy during fracture is changed. A new crack dissipation functional is established using the energy equivalent approach. By introducing a novel rational degradation function, not only can the strength of material failure be effectively utilized, but the model can also reproduce the cohesive softening relationship. A multi-field finite element method is used to discretize the model governing equations, and the equations are solved by an efficient BFGS monolithic algorithm. Finally, some representative numerical examples are used to analyze the effects of parameters in degradation function, length scale and mesh size on the results. The presented numerical simulation results demonstrate length scale and mesh scale independence, and are in good agreement with the experimental results and previous numerical results. At the same time, the numerical results also exhibit cohesive softening properties similar to the current phase field cohesive zone model. These results verify the robustness and effectiveness of the modified phase field model presented in this paper for simulating quasi-brittle failure and brittle fracture. [ABSTRACT FROM AUTHOR]

Drop impact is common in industrial applications and nature. As in the 3D printing, the drops squeezed out of the nozzle do not have to take on a spherical shape. This paper delves into the role of drop shape in drop impact with solidification under the 3D printing conditions. To track the rapidly evolving liquid-gas interface, the Cahn-Hilliard-based phase field method is employed. The solid-liquid interface is tracked by the liquid fraction. The model is discretized using an explicit finite difference scheme on a half-staggered grid. The code was written in FORTRAN and enhanced with the OpenMP technique. After being validated against experimental data, the model was applied to various cases. Moreover, the paper took into account a number of factors influencing drop impact and solidification dynamics, such as aspect ratio, contact angle, and thermal contact resistance. It was found that the aspect ratio exerts little effect on drop profile for the cases under consideration when solidification is on, but that the contact angle could drastically reduce freezing time, converting the convex solidfication pattern to a concave one. [ABSTRACT FROM AUTHOR]

The frequency of the applied electric field and the epitaxial strain can affect the microstructure and the overall properties of ferroelectric thin films. In this study, the ferroelectric characteristics of BaTiO3 nano single crystal thin films were investigated via phase field simulation under the epitaxial strain of -0. 1% and -0. 7%, respectively, in the frequency range of 0. 1 ~100 kHz. It is found that, with the increase in frequency, the square-shaped hysteresis loop gradually changes to an elliptic loop and the butterfly loop changes to a kidney-shaped loop. When the applied frequency is below 50 kHz, the coercive field increases rapidly with the increase in frequency, but the change in remnant polarization is not significant. When the frequency is over 50 kHz, the coercive field only increases slightly, and the remnant polarization exhibits a downward trend. It is observed that the frequency dependence of ferroelectric properties is more significantly affected by the epitaxial strain while the frequency is low. The frequency dependence is not very sensitive to the epitaxial strain while the frequency is high. Meanwhile, the compressive epitaxial strain causes significant rise in both the remnant polarization and coercive field, and the tensile epitaxial strain leads to opposite effects. By analysis, it is found that the underlying mechanism of the frequency-dependent hysteresis is due to the competition between the speed of microstructure evolution and the frequency of the applied electric field. The findings of this study serve as theoretical basis for the experiment and design of ferroelectric functional thin films, and they provide knowledge-based support for the application of high-frequency electronic devices. [ABSTRACT FROM AUTHOR]

• Phase field fracture model capable of finite deformation simulation for flexible piezoelectric. • The coupled governing equations are solved via SPSA and RCSA-EO strategies. • Can predict the fracture behavior of flexible piezoelectrics under electromechanical environments. Fracture failure is a major concern in mechanical engineering, particularly for piezoelectric materials. In contrast to other numerical methods, phase field method has significant advantages in addressing fracture progress. It can automatically track crack surfaces through ordered parameter evolution, which is versatile for modeling complex fracture behaviors. However, previous studies on phase field fracture in piezoelectric solids have primarily focused on brittle ceramics with small deformation. In recent years, flexible piezoelectrics with high stretchability have already been achieved in industrial production. These materials exhibit obvious nonlinear characteristics during deformation, which renders the traditional assumption of small deformations inadequate for predicting their fracture behaviors. In this work, we propose a finite deformation phase field fracture model for flexible piezoelectric materials, building upon the established nonlinear electromechanical material model. The numerical framework is carried out in the commercial software ABAQUS via a user element subroutine. Both single–pass staggered algorithm (SPSA) and residual-controlled staggered algorithm with even-odd iteration split (RCSA-EO) are employed to solve coupled electro-mechanical-phase field governing equations. The proposed model is validated through comparisons with analytical solutions and existing literature. Moreover, the developed numerical framework effectively explains the nonlinear fracture behavior observed in experiments conducted on Polyvinylidene fluoride (PVDF), a flexible piezoelectric material with a large failure strain. Numerical simulations are also performed to demonstrate the influence of the applied electric field on electromechanical fracture behavior. The results highlight that the specific impact of electric fields depends on material parameters, geometric parameters, and boundary conditions. The developed model is capable of making accurate and realistic predictions of fracture in flexible piezoelectric materials. This is particularly important for evaluating the reliability and safety of flexible piezoelectric devices. [ABSTRACT FROM AUTHOR]

A framework of the topology optimization incorporated with the phase field method considering the interfacial damage for optimizing the fracture resistance of inclusion-matrix composites is presented. The topology optimization was performed to redistribute the inclusion phase in order to reduce its volume while keeping the fracture resistance value of the initial design unchanged. The phase field method uses two scalar phase field variables: one is for the bulk crack and the other is for the interfacial crack. The decomposition of the strain tensor into compression and tension parts was incorporated into this phase field method to improve the mechanical behaviors of the materials. These compression and tension strain parts are orthogonal in the context of the inner product in which the tensor of elastic stiffness behaves as a metric. Moreover, in the simulation process, an investigation of the effects of the interfacial parameters on the numerical results was discussed. Through the obtained results, the method proposed is demonstrated to be accurate and efficient in eliminating spurious effects and singularity points on the behavior curves in the damage process. [ABSTRACT FROM AUTHOR]

The 6–8 wt% yttria-stabilized zirconia with a tetragonal structure (t′-YSZ) is extensively employed in thermal barrier coatings. The exceptional fracture toughness of t′-YSZ can be attributed to its distinctive ferroelastic toughening mechanism. Micro-structure and interface tension play a critical role in ferroelastic variant switching at the micro- and nano-scale. This paper presents an original thermodynamically consistent phase field (PF) theory for analyzing ferroelastic variant switching at the micro- and nano-scale of t′-YSZ. The theory incorporates strain gradient elasticity using higher-order elastic energy and interface tension tensor via geometric nonlinearity to represent biaxial tension resulting from interface energy. Subsequently, a mixed-type formulation is employed to implement the higher-order theory through the finite element method. For an interface in equilibrium, the effects of strain gradient elasticity result in a more uniform distribution of stresses, whereas the presence of interface tension tensor significantly amplifies the stress magnitude at the interface. The introduction of an interface tension tensor increases the maximum value of stress at the interface by a factor of 4 to 10. The nucleation and evolution of variants at a pre-existing crack tip in a mono-phase t′-YSZ have also been studied. The strain gradient elasticity is capable of capturing the size effect of ferroelastic variant switching associated with microstructures in experiments. Specifically, when the grain size approaches that of the specimen, the critical load required for variant switching at the crack tip increases, resulting in greater dissipation of elastic energy during ferroelastic variant switching. Moreover, the interface tension accelerates the evolution of variants. The presented framework exhibits significant potential in modeling ferroelastic variant switching at the micro- and nano-scale. [ABSTRACT FROM AUTHOR]

Purpose: This study aimed to explore how the microarchitectural features of lacunae and perilacunar zones impact the biomechanics of microdamage accumulation in cortical bone, crucial for understanding bone disorders' pathogenesis and developing preventive measures. Methods: Utilizing the phase field finite element method, the study analyzed three bone unit models with varying microarchitecture: one without lacunae, one with lacunae and one including perilacunar zones, to assess their effects on cortical bone's biomechanical properties. Results: The presence of lacunae was found to increase microcrack initiation risk, acting as nucleation points and accelerating microcrack propagation. Proximity to Haversian canals exacerbated stress concentration, speeding microdamage progression. Conversely, perilacunar zones mitigated both initiation and propagation. An elevated critical energy release rate correlated with slower crack growth and reduced damage severity. Conclusions: The research sheds light on the intricate mechanisms governing microcrack behavior in compact bone, highlighting the significant role of bone's microarchitectural features in its biomechanical response to microdamage. These insights are valuable for the development of strategies to prevent and treat bone-related disorders. [ABSTRACT FROM AUTHOR]

Copyright of Inorganic Chemicals Industry is the property of Editorial Office of Inorganic Chemicals Industry 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.)

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.

The evolution of interfaces is intrinsic to many physical processes ranging from cavitation in fluids to recrystallization in solids. Computational modeling of interface motion entails a number of challenges, many of which are related to the range of topological transitions that can occur over the course of the simulation. Microstructure evolution in a polycrystalline material that involves grain boundary motion is a particularly complex example due to the extreme variety, heterogeneity, and anisotropy of grain boundary properties. Accurately modeling this process is essential to determining processing-structure–property relationships in polycrystalline materials though. Simulations of microstructure evolution in such materials often use diffuse interface methods like the phase field method that are advantageous for their versatility and ease of handling complex geometries but can be prohibitively expensive due to the need for high interface resolution. Discrete interface methods require fewer grid points and can consequently exhibit better performance but have received comparatively little attention, perhaps due to the difficulties of maintaining the mesh and consistently implementing topological transitions on the grain boundary network. This work explicitly compares a recently-developed discrete interface method to a multiphase field method on several classical problems relating to microstructure evolution in polycrystalline materials: a shrinking spherical grain, the steady-state triple junction dihedral angle, and the steady-state quadruple point dihedral angle. In each case, the discrete method is found to meet or outperform the multiphase field method with respect to accuracy for comparable levels of refinement, demonstrating its potential efficacy as a numerical approach for microstructure evolution in polycrystalline materials.

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.

AbstractThe performance of the gas diffusion layer (GDL) plays a pivotal role in ensuring the efficient and stable operation of proton exchange membrane fuel cells (PEMFCs). Microstructure of the GDL significantly influences its internal water transport capabilities. In this article, the lattice array method is employed to construct ordered microstructures within GDLs, followed by the utilization of phase field method to numerically simulate water transport among these structures. The effects of lattice edge length, lattice height, and lattice shape on water transport throughout the GDL are investigated. The results reveal that excessively large or small lattice edge lengths detrimentally affect effective water transport within the GDL. Additionally, a smaller lattice height can accelerate the breakthrough of liquid water through the GDL. Notably, when the lattice height is smaller than the lattice edge length, it enhances the mass transfer. Moreover, the triangular lattice impedes the water flow, while the rectangular structure diminishes the transmission efficiency of liquid water. In contrast, hexagonal and octagonal structures exhibit the ability to alleviate the transport resistance of liquid water, with the octagonal lattice demonstrating superior mass transfer effectiveness. These findings underscore the paramount importance of meticulous structural design in further augmenting the mass transfer performance of ordered GDLs. The research provides meaningful guidance for the design of GDL structures. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of turbulence and interfacial waves in an oil–water channel flow. We consider a stratified configuration, in which a thin layer of oil flows on top of a thick layer of water. The oil–water interface that separates the two layers mutually interacts with the surrounding flow field, and is characterized by the formation and propagation of interfacial waves. We perform direct numerical simulation of the Navier-Stokes equations coupled with a phase field method to describe the interface dynamics. For a given shear Reynolds number, R e τ = 300 , and Weber number, W e = 0.5 , we consider three different types of oils, characterized by different viscosities, and thus different oil-to-water viscosity ratios μ r = μ o / μ w (being μ o and μ w oil and water viscosities). Starting from a matched viscosity case, μ r = 1 , we increase the oil-to-water viscosity ratio up to μ r = 100 . By increasing μ r , we observe significant changes both in turbulence and in the dynamics of the oil–water interface. In particular, the large viscosity of oil controls the flow regime in the thin oil layer, as well as the turbulence activity in the thick water layer, with direct consequences on the overall channel flow rate, which decreases when the oil viscosity is increased. Correspondingly, we observe remarkable changes in the dynamics of waves that propagate at the oil–water interface. In particular, increasing the viscosity ratio from μ r = 1 to μ r = 100 , waves change from a two-dimensional, nearly-isotropic pattern, to an almost monochromatic one. [ABSTRACT FROM AUTHOR]

In this article, we consider a gradient flow stemming from a problem in two-scale topology optimization. We use the phase-field method, where a Ginzburg–Landau term with obstacle potential is added to the cost functional, which contains the usual compliance but also an additional contribution including a local volume constraint in a penalty term. The minimization of such an energy by its gradient-flow is analyzed in this paper. We use an regularization and discretization of the associated state-variable to show the existence of weak solutions to the considered system. [ABSTRACT FROM AUTHOR]

Copyright of Petroleum Geology & Recovery Efficiency is the property of Petroleum Geology & Recovery Efficiency 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.)

Stiffness degradation resulting from matrix cracking has a great impact on the vibrational behaviors of composite laminates. However, the vibration frequencies of functionally graded graphene-reinforced composite (FG-GRC) plates with crack defects remain largely unexplored. In this paper, the variational phase field theory is implemented to investigate the vibration characteristics of cracked FG-GRC plates subjected to thermal conditions. The effective elastic constants depending on the temperature of FG-GRC plates are characterized by the extended Halpin-Tsai model. The first-order shear deformation theory (FSDT) is employed to describe the displacement fields of the cracked FG-GRC plates. The meshfree kernel particle method is engaged to discretize governing equations over the computational domain and then the obtained governing eigen-equations are calculated using the Ritz methodology. The numerical results illustrate the influences of crack length, crack number, the strength of foundations, graphene distribution, temperature variation and geometric values on the vibration fundamental frequencies of the FG-GRC plates. The present work not only provides an alternative and feasible approach for solving the interaction problems between cracking and vibration, but also makes significant advances to characterize the vibrational behaviors of cracked functionally graded materials for engineering applications. [ABSTRACT FROM AUTHOR]

Topology optimization of eigenfrequencies has significant applications in science, engineering, and industry. Eigenvalue problems as constraints of optimization with partial differential equations are solved repeatedly during optimization and design process. The nonlinearity of the eigenvalue problem leads to expensive numerical solvers and thus requires huge computational costs for the whole optimization process. In this paper, we propose a simple yet efficient linearization approach and use a phase field method for topology optimization of eigenvalue problems with applications in two models: vibrating structures and photonic crystals. More specifically, the eigenvalue problem is replaced by a linear source problem every few optimization steps for saving computational costs. Numerical evidence suggests first-order accuracy of approximate eigenvalues and eigenfunctions with respect to the time step and mesh size. Numerical examples are presented to illustrate the effectiveness and efficiency of the algorithms. [ABSTRACT FROM AUTHOR]

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.

Laser cladding is a new additive manufacturing technology that has emerged in recent years. It offers the advantages of a dense cladding layer structure and good bonding with the substrate, making it suitable for repairing parts surfaces and with broad application prospects. Mechanical properties are key performance indicators of the cladding layer. However, the small size and inhomogeneous structure of the cladding layer make it difficult to prepare mechanical properties tests. The microstructure is a key factor affecting the mechanical properties of the cladding layer. Therefore, it is urgent to carry out research on microstructure simulation of the laser cladding layer in order to obtain its mechanical properties. This paper provides a summary of the commonly used microstructure simulation methods, including the research achievements of phase field method, Monte Carlo (MC) method, and cellular automaton (CA) method in the microstructural simulation of laser cladding and metal solidification. The existing problems are analyzed, and the future development direction of this field is forecasted. The phase field method has high accuracy in its simulation results and does not require complex solid–liquid interface tracking, but its solution is complex, and its calculation efficiency is low due to the large calculation amount. The MC method is relatively simple to calculate and does not assume specific grains, but lacks physical basis for the quantitative analysis of the impact of various physical phenomena. The CA method is a commonly used microstructure simulation method with clear physical significance, high computational efficiency, and low cost and has broad prospects for development. [ABSTRACT FROM AUTHOR]

To provide reference for controlling the generation of uniform high frequency of droplets, aimed at droplet generation in confined microchannels, considering the influence of fluid physical properties and flow conditions, numerical simulation of droplet formation in a T-confined microchannel was carried out. Using COMSOL software, droplet generation process simulation model was established, and the phase field method was adopted to explore the capillary number, two-phase flow ratio, viscosity ratio and wall wettability effects on droplet generation. Results showed that when the continuous phase viscosity is 0.01 Pa·s, and the dispersed phase velocity and viscosity are 0.005 m/s and 0.001 Pa·s, respectively, as the continuous phase capillary number increases from 0.001 5 to 0.007 5, the continuous phase capillary number increased from 0.001 5 to 0.007 5, the dimensionless droplet length decreased from 2.8 power index to 1.5, the dimensionless liquid column length increased linearly from 2.6 to 7.6, the droplet generation frequency increased from 30 power exponents per second to 52; when the continuous phase viscosity is 0.01 Pa·s, the dispersed phase viscosity is 0.001 Pa·s, and the flow ratio increased from 0.1 to 0.9, the dimensionless droplet length increased linearly from 1.05 to 2.55, the dimensionless liquid column length decreased from 6.49 power index to 2.51, the droplet generation frequency decreased from 170 power index per second to 65, when the continuous phase velocity is 0.01 m/s, the dispersed phase velocity is 0.005 m/s, and the viscosity ratio increased from 0.025 to 0.5, and the dimensionless droplet length increased from 1.72 to 2.1 The length of the dimensionless liquid column increased from 3.11 to 3.9, and the droplet generation frequency decreased from 31 functions per second to 24. Wall wettability affects the morphology of droplets, but there is no obvious functional relationship. The physical properties and flow conditions of fluid have significant influence on droplet formation. [ABSTRACT FROM AUTHOR]

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]

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.

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]

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]