Phase-field Model Research Articles

A phase-field framework for modeling multiple cohesive fracture behaviors in laminated composite materials

This work proposes a phase-field (PF) framework to characterize the multiple fracture behaviors and quasi-brittle failure mechanisms in fiber reinforced laminated composites. The multi-phase field model considering the distinct failure mechanisms of composites is first derived based on thermodynamic principles. A hybrid PF formulation is proposed through the reasonable definition of the energy functional for each field, and a PF regularized cohesive zone model (CZM) combined with Hashin failure criteria is achieved for the typical failure modes in composites. The proposed PF-CZM is endowed with a linear softening behavior and is proven to be independent of the internal length scale parameter. The multiple fracture modeling of composite laminates is achieved by coupling the PF model for intralaminar failure with the cohesive interface element for interlaminar one. The model’s validation is demonstrated by several representative numerical examples, and results show that both the intralaminar fiber/matrix fractures and the induced interlaminar delamination, as well as their complex interactions, are well captured by the proposed model.

Aiming the water transport issues of anion exchange membrane fuel cells by introducing hydrophobic carbon nanotube diffusion layer

The gas diffusion layer (GDL) enhances the transport efficiency of reaction gases and facilitates the removal of accumulated liquid water, thereby improving water management in anion exchange membrane fuel cells (AEMFCs). Carbon nanotubes (CNTs) are recognized as one of the promising materials to optimize mass transport within GDLs. In this study, we successfully synthesized an abundance of CNTs on the surface of commercial gas diffusion media. The as-prepared CNT layer exhibits exceptionally high hydrophobic properties and forms a hierarchically hydrophobic structure combined with the gas diffusion media, effectively enhancing the mass transfer of the membrane electrode assembly (MEA) and reducing ohmic impedance. Under supersaturation conditions, the power density of the MEA containing CNTs reaches 1031.7 mW/cm2, significantly higher than the 760.64 mW/cm2 observed for the commercial GDL. Further testing demonstrates that the MEA containing CNTs exhibits a limiting current density of 2507.4 mA/cm2, which is much superior to the benchmark MEA with the commercial GDL (1591 mA/cm2). Electrochemical impedance spectroscopy analysis reveals that the mass transfer resistance of MEAs with CNTs is lower than that of MEAs with the commercial GDL. More importantly, the phase-field modeling is performed to simulate the transport of the liquid water in the hierarchical GDL.

Cohesive phase-field model for dynamic fractures in coal seams

A novel rate-dependent cohesive phase-field model that can simulate fluid-driven dynamic quasibrittle fracture in soft coal rocks is presented. The coal matrix and fractures are described by a unified fluid continuity equation, and the Poiseuille flow in the fractures is modeled by deformation-dependent permeability. Dynamic evolution equations for porosity, permeability and the Biot coefficient during phase-field evolution are developed. The phase-field evolution equation is derived from the total energy generalization based on the Francfort–Marigo variational principle. To simulate the rate-dependent dynamic fracture problem accurately, the effects of the phase-field evolution dissipation energy and inertia energy are considered. The above multifield coupled system is solved via a staggered approach within the finite element framework via the Newton–Raphson iterative algorithm. The independence and convergence of the proposed model were verified, and its accuracy was validated via an analytical solution for crack width and a benchmark test for dynamic crack propagation. The propagation law of single/multicluster hydraulic fractures in coal seams with interlayers was investigated via the proposed model. The simulations revealed that the number of clusters and the mechanical properties of the interlayers significantly influence the fracture morphology. Finally, hydraulic fracturing of fracture-developed coal seams was simulated, confirming the potential of the proposed model.

Flexoelectric influence on crack propagation in ferroelectric single crystals: a phase-field approach

Ferroelectric materials, known for their inherent brittleness, are prone to brittle fracture. This limitation not only curtails the materials’ operational lifespan but also impinges on the reliability of the associated devices. Notably, the pronounced strain gradient present at the crack tip necessitates consideration of the flexoelectric effect—the interaction between strain gradient and electric polarization—in the fracture mechanics of ferroelectric materials. This study introduces a phase-field model incorporating the flexoelectric effect to elucidate its role on crack growth and domain evolution in ferroelectric single crystals. Our findings demonstrate that both crack trajectory and domain switching phenomena at the crack’s forefront are substantially influenced by the magnitude and sign of the flexoelectric coefficient, as well as the initial polarization direction. Depending on the computational scenarios, the flexoelectric effect can either exacerbate or impede crack propagation. Through meticulous examination of the mechanical field distributions and their temporal progression, we have uncovered the underlying mechanisms by which the flexoelectric effect governs crack propagation in ferroelectric single crystals. These insights pave the way for improving the fracture resistance and thereby enhancing the reliability of ferroelectric devices.

A three-dimensional coupled thermo-elastic-plastic phase field model for the brittle-ductile failure mode transition of metals

Dynamic brittle fracture and shear banding are two typical failure modes of metals, and the transformation of the brittle-ductile failure mode has been observed in the Kalthoff test. This paper establishes a thermo-elastic-plastic coupled three-dimensional phase field model to simulate brittle-ductile failure mode transition of metals. The expression for the variation of the Taylor-Quinney coefficient with stress triaxiality is adopted, and the critical energy release rate is automatically adjusted using the Taylor-Quinney coefficient. Then, the Kalthoff test is simulated using the proposed model. The brittle-ductile failure mode transformation phenomenon is reproduced, which agrees well with the experimental results. It can be well proved that impact velocity is crucial in determining the transition to failure mode. At low-velocity impact, the energy is insufficient to drive the plastic accumulation of the shear band, resulting in brittle tensile fracture. At high-velocity impact, the energy is sufficient to drive the formation of adiabatic shear bands, resulting in tensile shear failure. In addition, three-dimensional simulations show that the tip of the shear band exhibits a crescent-shaped non-two-dimensional extension state under finite thickness. This numerical framework provides a predictive tool to understand the evolution of the dynamic failure of metals under impact loading.

Modeling the electro-chemo-mechanical failure at the lithium-solid electrolyte interface: Void evolution and lithium penetration

The solid-solid contact interface is crucial for the reliability of solid-state energy storage systems. The contact condition becomes more complicated when lithium (Li) metal is used as the anode. The contact between solid electrolyte (SE) and Li metal is inferior compared to the liquid/solid interface in conventional Li-ion batteries. Experimental evidence has shown that improper operating conditions of solid-state batteries can lead to electro-chemo-mechanical failures at the Li/SE interface, including the formation of voids and the penetration of Li. In this study, a unified phase-field model is developed to investigate these two mechanisms. The model considers the coupled electro-chemo-mechanical processes including void diffusion, lattice annihilation, stripping and plating reactions, and plastic deformation of Li metal. The study begins with a revisit of the deformation-mechanism map for Li metal under a wide range of temperatures, stress, and deformation rates. This map serves as the basis for the mechanical characterization in the phase-field model. The large inelastic deformation of Li is considered by introducing an advection term into the Allen-Cahn equation, which is used to describe the dynamic evolution of the Li and void phases. The effects of current density and stack pressure on void evolution and Li penetration are studied based on the model predictions. By combining the simulation results with the experimental data from publications, we obtain the stable operation zone of stack pressure and applied current density. In this zone, the Li/SE interface can enable stable stripping and plating of Li metal. The same phase-field modeling framework is transferred to investigate the Li-Mg alloy/SE interface considering Li-Mg alloy is also used as the anode. The fundamental difference between Li/SE and Li-Mg/SE is analyzed accordingly. This study provides a useful tool for the design, manufacturing, and management of next-generation batteries by providing important scientific insights into the electro-chemo-mechanical processes of different anode materials under various operational conditions.

Analysis and numerical methods for nonlocal‐in‐time Allen‐Cahn equation

AbstractIn this paper, we investigate the nonlocal‐in‐time Allen‐Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well‐posedness of the NiTACE by establishing the maximal regularity for the nonlocal‐in‐time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy‐stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method.

A class of efficient high-order time-stepping methods for the anisotropic phase-field dendritic crystal growth model

In this paper, we develop and analyze a novel class of arbitrarily high-order and unconditionally energy stable schemes for the anisotropic phase-field dendritic crystal growth model, which is a highly nonlinear system that combines the anisotropic Allen–Cahn equation with the thermal equation. The proposed schemes are based on an extrapolated and linearized Runge–Kutta method for an auxiliary variable reformulation of the crystal growth model. A delicate implementation demonstrates that the proposed method can be realized in a very efficient way, requiring only the solution of a coupled linear elliptic system at each time step. We prove that the constructed schemes satisfy the energy dissipation property and demonstrate the convergence order through a consistency error analysis. To the best of our knowledge, this is the first unconditionally stable scheme of arbitrarily high-order for the anisotropic phase-field dendritic crystal growth model. Numerical experiments for two and three dimensional dendritic crystal growth are simulated to verify our theoretical accuracy as well as the efficiency of the proposed method. Furthermore, to emphasize the superiority of the high-order schemes, we present a numerical comparison that directly contrasts its performance with the lower-order schemes.

Isogeometric collocation method to simulate phase-field crystal model

PurposeThe purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.Design/methodology/approachFirst, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.FindingsIn the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.Originality/valueThis research work introduces the IGC method as a simulation technique for the phase-field crystal model.

In situ evaluation and manipulation of lithium plating morphology enabling safe and long‐life lithium‐ion batteries

AbstractThe morphology of plated lithium (MPL) metal on graphite anodes, traditionally described as “moss‐like” and “dendrite‐like”, exert a substantial negative influence on the performance of lithium‐ion batteries (LIBs) by modulating the metal‐electrolyte interface and side reaction rates. However, a systematic and quantitative analysis of MPL is lacking, impeding effective evaluation and manipulation of this detrimental issue. In this study, we transition from a qualitative analysis to a quantitative one by conducting a detailed examination of the MPL. Our findings reveal that slender lithium dendrites reduces the lifespan and safety of LIB by increasing the side reaction rates and promoting the formation of dead lithium. To further evaluate the extent of the detrimental effect of MPL, we propose the specific surface area (SSA) as a critical metric, and develop an in situ method integrating expansion force and electrochemical impedance spectroscopy to estimate SSA. Finally, we introduce a pulse current protocol to manipulate hazardous MLP. Phase field model simulations and experiments demonstrate that this protocol significantly enhances the reversibility of plated lithium. This research offers a novel morphological perspective on lithium plating, providing a more detailed fundamental understanding that facilitates effective evaluation and manipulation of plated lithium, thereby enhancing the safety and extending the cycle life of LIBs.image

Dynamic swarms regulate the morphology and distribution of soft membrane domains.

We study the dynamic structure of lipid domain inclusions embedded within a phase-separated reconstituted lipid bilayer in contact with a swarming flow of gliding filamentous actin. Passive circular domains transition into highly deformed morphologies that continuously elongate, rotate, and pinch off into smaller fragments, leading to a dynamic steady state with ≈23× speedup in the relaxation of the intermediate scattering function compared with passive membrane domains driven by purely thermal forces. To corroborate experimental results, we develop a phase-field model of the lipid domains with two-way coupling to the Toner-Tu equations. We report phase domains that become entrained in the chaotic eddy patterns, with oscillating waves of domains that correlate with the dominant wavelengths of the Toner-Tu flow fields.

A consistent phase-field model for three-phase flows with cylindrical/spherical interfaces

A phase-field model for three-phase flows with cylindrical/spherical interfaces is established by combining the Navier-Stokes (NS), the continuity, and the energy equations, with an explicit form of curvature-dependent modified Allen-Cahn (AC) and Cahn-Hilliard (CH) equations. These modified AC and CH equations are proposed to solve the inconsistency of the phase-field method between flat and curved interfaces, which can result in “phase-vanishing” problems and the break of mass conservation during the phase-changing process. It is proved that the proposed model satisfies the energy dissipation law (energy stability). Then the icing process with three phases, i.e., air, water, and ice, is simulated on the surface of a cylinder and a sphere, respectively. It is demonstrated that the modification of the AC and CH equations remedies the inconsistency between flat and curved interfaces and the corresponding “phase-vanishing” problem. The evolution of the curved water-air and the water-ice interfaces are captured simultaneously, and the volume expansion during the solidification owing to the density difference between water and ice agrees with the theoretical results. A two-dimensional icing case with bubbles rising is simulated. The movement and deformation of bubbles, as well as the evolution of the interfaces, effectively illustrate the complex interactions between different phases in the icing process with phase changes.

Phase field study of pitting corrosion: Electrochemical reactions and temperature dependence

This study presents a new phase field model of pitting corrosion for metal materials, encapsulating the transport of ionic species and electrochemical reactions in the electrolyte, and the effect of temperature on corrosion kinetics is incorporated by relating the interface kinetics parameter to the temperature. The Allen-Cahn and Cahn-Hilliard equations, along with the Poisson’s equation, is established to govern phase transformation, ions diffusion and electrostatic potential distribution, respectively. The model is validated by conducting several benchmark numerical studies. Simulation results demonstrate that within the temperature range of 10 °C to 20 °C, the model closely matches experimental data and result from sharp interface model. Additionally, the model effectively reproduces the pit shapes observed in experiments and captures variations in ion concentrations within the electrolyte. Several examples of the model simulating corrosion problems with complex evolving morphologies are provided.

Fracture modeling of CNT/epoxy nanocomposites based on phase-field method using multiscale strategy

The computational modeling of fracture, particularly in structures with complex crack topologies, remains challenging due to significant computational costs, especially in simulating two- and three-dimensional brittle fracture. This study presents an efficient phase-field model to address these challenges. By leveraging the user (UMAT) subroutine in ABAQUS and establishing an analogy between the phase-field evolution law and the heat transfer equation, the method efficiently tackles complex fracture problems. The model is verified through analysis of typical 2D and 3D fracture benchmarks with different failure modes, demonstrating accuracy and efficiency compared to experimental and numerical data. Additionally, the model is applied to explore brittle fracture in carbon nanotubes (CNTs)/epoxy nanocomposites, revealing insights into the impact of CNT weight fraction on fracture phenomena prediction. The incorporated CNTs in the matrix are considered uniformly dispersed and randomly oriented. Overall, the developed model and computational implementation show promise for meeting the requirements of structural-level engineering practices.

An integrated modeling framework with open architecture for phase field simulation of multi-component alloys

An integrated modeling framework (PanPhaseField) has been developed, which enables a direct and fast coupling between CALPHAD calculations and large-scale phase field simulations for multi-component alloys. It adopts an open architecture allowing for integration of user-defined phase field models in a plug-and-play manner by taking full advantage of the user-friendly graphical interface of Pandat software. The developed modeling platform becomes an enabling tool that can be used to simulate the evolution of spatially varying microstructures of industrial complex alloys for various engineering applications.

Calibration and validation of a phase-field model of brittle fracture within the damage mechanics challenge

In the context of the Damage Mechanics Challenge, we adopt a phase-field model of brittle fracture to blindly predict the behavior up to failure of a notched three-point-bending specimen loaded under mixed-mode conditions. The beam is additively manufactured using a geo-architected gypsum based on the combination of bassanite and a water-based binder. The calibration of the material parameters involved in the model is based on a set of available independent experimental tests and on a two-stage procedure. In the first stage an estimate of most of the elastic parameters is obtained, whereas the remaining parameters are optimized in the second stage to minimize the discrepancy between the numerical predictions and a set of experimental results on notched three-point-bending beams. The good agreement between numerical predictions and experimental results in terms of load–displacement curves and crack paths demonstrates the predictive ability of the model and the reliability of the calibration procedure.

High-throughput phase field simulation and machine learning for predicting the breakdown performance of all-organic composites

All-organic dielectric polymers are materials of choice for modern power electronics and high-density energy storage, and their performance can be significantly improved by doping trace amounts of organic molecular semiconductors with strong electron-affinity energy to suppress charge conduction losses. Insight into the breakdown mechanism of polymers/organic molecular semiconductor composites is essential for the design of high-performance dielectric polymers. This study investigates the impact of the doping concentration of organic molecular semiconductors, dielectric constants, and trap depths on the breakdown performance of dielectric polymers under high temperature and electric fields. A modified phase-field model, incorporating deep traps and carriers’ coulomb capture radius, has been developed to facilitate high-throughput simulations of electrical breakdown in polymer/organic molecular semiconductor composites. This work accurately predicted the breakdown strength of all-organic composites using high-throughput phase-field simulation data as input for machine learning, which provides crucial theoretical support for designing all-organic composite dielectric polymers for energy storage capacitors under extreme conditions.

Phase-field based modeling and simulation for selective laser melting techniques in additive manufacturing

In this study, we develop a phase-field model to describe the solid–liquid phase changes, heat conduction phenomena, during the selective laser melting process. This model is based on the variational principle of minimizing the free energy functional. The proposed model integrates the phase-field equation and the energy equation, which are used to capture the dynamical behavior of the interfacial evolution. We use the semi-implicit Crank–Nicolson scheme and central difference to ensure second-order accuracy in time and space. The numerical scheme is unconditional energy stable. This paper rigorously proves the energy stability of the phase-field model of the Selective Laser Melting process, which confirms the numerical stability and the physical rationality of the solution. Various numerical experiments are performed to verify the robustness of our proposal model. This model can effectively simulate the energy transfer and shape structure changes of the products during the selective laser melting manufacturing process, which provides a reliable guarantee for predicting and optimizing the quality and performance of the selective laser melting process additive manufacturing process.

Employing Williams’ series for the identification of fracture mechanics parameters from phase-field simulations

Fracture mechanics and damage mechanics are two theories that describe the degradation of the bearing capacity of structures. Fracture mechanics is based on a discontinuous description of cracking, while damage mechanics proposes a continuous description of material degradation. These two approaches are often opposed in the literature, from both theoretical and numerical points of view. This work suggests correlating the two approaches by applying Williams’ series, usually dedicated to experimental results, to phase-field computations. Williams’ series are employed to extract equivalent fracture mechanics parameters as a post-processing step. The proposed analysis based on a fracture mechanics description excludes the fracture process zone. Typical fracture mechanics parameters such as energy release rate, stress intensity factors, fracture process zone size, and crack tip position are determined from the phase-field computations. The approach is illustrated on a two-dimensional structure representing a beam whose notch opening displacement is controlled. The dependence on the choice of the internal length of the phase-field model is studied. Similarities and differences between both modeling routes are discussed.

Kinetic Simulation of Turbulent Multifluid Flows

Despite its visual appeal, the simulation of separated multiphase flows (i.e., streams of fluids separated by interfaces) faces numerous challenges in accurately reproducing complex behaviors such as guggling, wetting, or bubbling. These difficulties are especially pronounced for high Reynolds numbers and large density variations between fluids, most likely explaining why they have received comparatively little attention in Computer Graphics compared to single- or two-phase flows. In this paper, we present a full LBM solver for multifluid simulation. We derive a conservative phase field model with which the spatial presence of each fluid or phase is encoded to allow for the simulation of miscible, immiscible and even partially-miscible fluids, while the temporal evolution of the phases is performed using a D3Q7 lattice-Boltzmann discretization. The velocity field, handled through the recent high-order moment-encoded LBM (HOME-LBM) framework to minimize its memory footprint, is simulated via a velocity-based distribution stored on a D3Q27 or D3Q19 discretization to offer accuracy and stability to large density ratios even in turbulent scenarios, while coupling with the phases through pressure, viscosity, and interfacial forces is achieved by leveraging the diffuse encoding of interfaces. The resulting solver addresses a number of limitations of kinetic methods in both computational fluid dynamics and computer graphics: it offers a fast, accurate, and low-memory fluid solver enabling efficient turbulent multiphase simulations free of the typical oscillatory pressure behavior near boundaries. We present several numerical benchmarks, examples and comparisons of multiphase flows to demonstrate our solver's visual complexity, accuracy, and realism.