Phase-field Model Research Articles

A crystal plasticity phase-field model for microstructure sensitive fatigue crack growth in a superalloy

In this work, a dislocation density-based crystal plasticity phase-field model (CP-PFM) is developed to simulate fatigue crack growth in nickel-based superalloys. Through normalization validation, the plastic dissipation work and crystallographic work are shown to be consistent with the fatigue indicator factors (FIPs), cumulative equivalent plastic strain and cumulative shear strain, and the two energies are computed as the main driving forces of the phase field. Both driving force models are able to obtain fatigue crack growth in close approximation to the experimental rate. However, the model with crystallographic work as the main driving force obtains crack growth paths that are in better agreement with electron backscattering pattern (EBSD) observations, which is attributed to its greater ability to characterize the microstructural susceptibility of fatigue crack growth. Specifically, the model is able to capture the tendency of cracks to crack along the close-packed planes and the hindering effect of grains with large misorientation angles on fatigue crack growth, which together contribute to the curved morphology of fatigue cracks. The combination of large grains or grains with small misorientation angles favors persistent slip band (PSB) formation and leads to softening of the crack tip, which results in lower fatigue crack growth rates.

Cracking Mechanism and Inhibition Strategies of Polycrystalline NCM Electrode Particles.

Developing a high-energy-density cathode material (LiNi1-x-yCoxMnyO2, NCM) for lithium-ion batteries is crucial to the electric vehicle and energy storage industries. However, the continuous insertion/extraction of Li+ generates diffusion-induced stress, causing NCM particles to crack or even pulverize, leading to battery capacity loss and limiting its wider commercial application. Current experimental studies are primarily postmortem examinations, and it is difficult to capture the particle cracking evolution. Simulation studies frequently ignore or simplify anisotropic volume contraction, demonstrating an insufficient understanding of the cracking mechanism of NCM polycrystalline particles, and cracking prevention strategies still need improvement. Therefore, we develop an anisotropic polycrystalline fracture phase-field model (AP-FPFM) that focuses on the anisotropic volume contraction of primary particles and precisely generates grain boundary distribution, coupling with Li+ diffusion, mechanical stress, and particle cracking. We employ AP-FPFM to demonstrate the behavior and mechanism of NCM polycrystalline particle cracking and illustrate the necessity and importance of anisotropic volume contraction to understand particle cracking. Furthermore, we explore the effects of average primary particle size, secondary particle size, and core-shell structure modulation on crack initiation and propagation and propose strategies to inhibit or migrate NCM polycrystalline particle cracking. This work provides theoretical support for revealing the cracking mechanism of anisotropic polycrystalline NCM particles and supplying optimization strategies to suppress particle cracking and improve the mechanical stability.

Effect of grain size and grain boundary on the energy storage performance of polycrystalline ferroelectrics

Dielectric capacitors based on polycrystalline ferroelectrics have attracted much attention due to their significant power density and fast charge–discharge speed. The energy storage performance of polycrystalline ferroelectrics is highly dependent on the grain size and grain boundary. Here, the effect of grain size and grain boundary on the domain structures and polarization–electric field (P–E) hysteresis loops of polycrystalline ferroelectrics are investigated by using a phase-field model based on the time dependent Ginzburg–Landau (TDGL) equation. It is found that the depolarization field in the grain boundary induces the vortex domain when the grain size is reduced or the grain boundary thickness increases in certain extent, resulting in slender P–E loops, which contributes to an improvement in the energy storage efficiency and density. However, as the grain size further decreases or the grain boundary thickness further increases, the energy storage density decreases, which is attributed to the concurrent reduction in both the remnant and saturation polarizations. This study provides a considerable insight for optimizing the energy storage performance by carefully adjusting the grain size and grain boundary thickness in polycrystalline ferroelectrics.

Fracture simulation of fiber reinforced composite panels with holes

Fiber reinforced composite (FRC) with holes have broad applications in various fields. In this study, the influence of fiber orientation and hole distribution on the fracture behavior of FRC was investigated. A phase-field modeling was established to simulate the fracture process of the composite, and the mechanical performance of unidirectional fiber reinforced composite and woven fiber reinforced composite were analyzed, respectively. Our numerical results showed that fiber orientation and hole distribution have a significant impact on the fracture behavior of FRC. We observed that aligning the fibers parallel to the loading direction led to an increase in the maximum load bearing capacity of the composite. A more uniform hole distribution can enhance the overall mechanical performance of FRC. Furthermore, in the presence of thermal shock, crack propagation tends to grow towards the hole. These findings are of great significance for understanding the fracture behavior of FRC, and for optimizing material design and fabrication processes.

From process to property: multi-physics modeling of dislocation dynamics and microscale damage in metal additive manufacturing

AbstractMetal additive manufacturing (AM) has the potential to tailor the mechanical performance of materials. Due to the complex thermal history and unique microstructure, AM materials are reported to contain distinct dislocation networks with a high dislocation density, which affect the plastic deformation behavior and fracture. However, it is challenging to experimentally observe the formation of such dislocation structures. In this work, a multi-scale multi-physics crystal plasticity modeling framework that integrates the process-structure-property relationship in metal AM is developed. The temperature field obtained from thermal-fluid flow simulations of the AM process and the microstructure from the phase field model of grain growth are combined into thermo-mechanical crystal plasticity simulations to obtain grain-scale thermal stresses. These stresses are used as input to simulate the evolution of dislocation structures within individual grains. Taking AM 316L stainless steel as the material of interest, the effect of initial dislocation configuration on the slip plane and cross-slip mechanism on the dislocation structure formation are investigated. Furthermore, a phase field damage model is implemented to study the initiation of microscale damage and their relationship with dislocation structures, which is a main novelty of this work. This modeling framework provides comprehensive simulations of all aspects of metal AM and offers insights into the dislocation mechanisms and damage formation at microscale in AM materials, which could be used to guide the manipulation of the mechanical properties of AM materials.

A simple and efficient solution scheme of coupling method between phase field regularized cohesive zone model and linear elastic model for fracture

To improve computational efficiency, in this work, a coupling scheme between a phase field regularized cohesive zone model and a linear elastic model is proposed. According to the crack pattern, the entire solution domain is partitioned into different subregions, the phase field regularized cohesive zone model is applied only in the subregion where the crack appears, the linear elastic model is considered in the remaining region, combining the characteristics and advantages of the two models to improve computational efficiency. Secondly, in the light of the similarity between the phase field model governing equation and the heat transfer equation, the analogous relationships for different coefficients are given, and the coupling model is implemented using an equivalent thermal coupling framework, avoiding post-processing of complex element. Finally, the validity of the presented model is verified by some typical examples. The results indicate that the coupling model can be used to study the complex fracture process under different failure modes. Its load displacement response, energy characteristic and crack pattern are in line with that of the pure phase field regularized cohesive zone model and are in agreement with experiment and literature results. Meanwhile, compared to pure phase field model, the proposed coupling model consumes lower time cost.

Phase-field modeling of facet growth of diamond using a modified anisotropic coefficient of interfacial energy

In the present work, a modified anisotropic coefficient of interfacial energy is used to simulate the anisotropic facet crystal growth along non-orthogonal directions by phase-field method. The modified anisotropic coefficient of interface anisotropy is simulated and evaluated against the theoretical prediction of equilibrium shape. The simulation results are compared between the modified form and classic form. Subsequently, the modified anisotropic coefficient is employed to simulate the Wulff construction of facet growth of diamond, encompassing both pristine diamond and diamond with reconstructed surfaces. The dynamics of diamond morphology and the reconstructed surface area and proportion are thoroughly evaluated, demonstrating agreement with experimental records. The effect of the reconstruction surface on the growth of diamond grain is analyzed in terms of the total energy reduction, and the range of interfacial energies for more stable diamond reconstructions is also summarized. By introducing the modified anisotropic coefficient, anisotropic facet growth can be described and the simulation results are verified both theoretically and experimentally.

Adaptive scaled boundary finite element method for hydrogen assisted cracking with phase field model

This study introduces a numerical framework for analysing hydrogen assisted cracking, employing the scaled boundary finite element method. This is the first instance where the scaled boundary finite element method is employed to model hydrogen embrittlement. The phase field model is utilized to simulate defects, complemented by adaptive meshing using polytree mesh. The ability of the scaled boundary finite element method to treat polygonal elements assists in mitigating the problem of hanging nodes that arises from polytree decomposition. The adaptive framework is developed to predict crack propagation using an initial unstructured quadrilateral mesh generated from any commercial software. The hydrogen atom concentration depends on the hydrostatic stress gradient, which is calculated by interpolating nodal hydrostatic stress with scaled boundary shape functions and taking the gradient. A staggered solution approach is adopted to concurrently tackle hydrogen transport, elasticity, and phase field equations. The methodology is validated using Mode-I edge notch specimens and analysing crack propagation from corrosion pits, with results demonstrating close agreement with existing data. Subsequently, the framework is extended to address more intricate scenarios, such as crack propagation in X-shaped plates and hydrogen transmission in tanks. As the last example, more advanced image based fracture analysis of weld structure is investigated. This example studies the possibility of analysing the hydrogen diffusion directly from TEM imagery. Moreover, we investigate how hydrogen concentration influences structural failure.

A Phase-Field Model for Wet Snow Metamorphism.

The microstructure of snow determines its fundamental properties such as mechanical strength, reflectivity, or thermo-hydraulic properties. Snow undergoes continuous microstructural changes due to local gradients in temperature, humidity, or curvature, in a process known as snow metamorphism. In this work, we focus on wet snow metamorphism, which occurs when the temperature is close to the melting point and involves phase transitions among liquid water, water vapor, and solid ice. We propose a pore-scale phase-field model that simultaneously captures the three relevant phase change phenomena: sublimation (deposition), evaporation (condensation), and melting (solidification). The phase-field formulation allows one to track the temperature evolution among the three phases and the water vapor concentration in the air. Our three-phase model recovers the corresponding two-phase transition model when one phase is not present in the system. 2D simulations of the model unveil the impact of humidity and temperature on the dynamics of wet snow metamorphism at the pore scale. We also explore the role of liquid melt content in controlling the dynamics of snow metamorphism in contrast to the dry regime before percolation onsets. The model can be readily extended to incorporate two-phase flow and may be the basis for investigating other problems involving water phase transitions in a vapor-solid-liquid system, such as airplane icing or thermal spray coating.

Deep operator network surrogate for phase-field modeling of metal grain growth during solidification

A deep operator network (DeepONet) has been constructed that generates accurate representations of phase-field model simulations for evolving two dimensional metal grain morphology growing from melt. These representations serve as lower resolution, computationally efficient stand-ins for quick parameter space exploration of solutions to the Allen–Cahn equations that dictate the phase-field model simulations. The experimental target for the phase-field model is a uranium casting system cooling a 434 g uranium charge from a maximum temperature of 1400 °C at an average rate of 30 °Cmin, traversing the crystallographic phases of the pure metal. Experimental parameters inform the phase-field model, whose higher resolution computational model solutions are used to train the DeepONet in a given parameter space with the aim of developing a faster, more efficient method for predicting the solidifying metal’s microstructure at different potential experimental values. The final DeepONet generates high accuracy, lower resolution predictions with cumulative relative approximation error over all timesteps of less than 0.5%, while ensuring solutions remain within physically feasible ranges. These relative error values are comparable with other state-of-the-art DeepONet models for microstructure evolution, while significantly reducing the amount of training data required. Training a convolutional neural network simultaneously with the DeepONet, enforcing realistic values at the complex metal grain boundaries, and mathematically encoding boundary conditions into the structure of the DeepONet improved prediction accuracy and computational efficiency over a standard DeepONet model.

Far-field reactivation of natural fractures by stress shadow effect

Hydraulic fracturing is the most important means to achieve the reservoir stimulation of tight formations like shale, where the reservoir permeability is enhanced after the interaction between hydraulic fracture (HF) and natural fractures (NFs). Apart from the widely known direct intersection behaviors between HF and NFs (i.e., the HF meets the NFs), the NFs may also be reactivated by the stress shadow effect of HF, but the failure mechanism and reactivation modes are still unclear. To address this, the far-field reactivation of NFs under the stress shadow effect is described based on a hybrid phase field model (PFM) with the frictional contact criterion, where the open or slip conditions can be quantitatively calculated using the Mohr-Coulomb yield function along the contact interface of NFs. The zone with reactivated NFs is divided into mode II and mixed mode dominated sub-zones respectively, and the phase diagram of reactivation behaviors is summarized as variations of the location and angle of the NF. The NF reactivation modes are discussed under different geological conditions (strength of NFs, distribution angle of NFs, and initial stress difference). It could be found that the higher initial stress ratio leads to more mode II reactivated NFs, and NFs strengths and angles influence both mode II and mixed mode zones notably. The NF reactivation tends to take place with lower strength, higher injection rate, higher initial stress ratio, and special angle (e.g., HF are perpendicular to NFs), during which the permeability enhancement and reservoir stimulation are evaluated by the phase field model.

A microstructural defect-orientation informed phase field model

Micro-cracks, micro-voids and other such defects, which typically coalesce to form macroscopic cracks, could be represented through incompatible, local rotations of material points, i.e. micro-regions. Specifically, based on a micropolar or Cosserat continuum-like hypothesis that requires attaching directors to material points, we track the evolving frame rotation and hence the microstructural orientation during quasi-brittle damage. We introduce a critical energy release rate incorporating the wryness tensor, which in turn is a function of the micro-rotation field and its gradient within the damaged region. The (pseudo) rotation field appears as additional degrees of freedom to describe a frame field, whose evolution is particularly significant within a diffused region of evolving damage as obtained through a phase-field formulation of brittle fracture. We emphasize that, unlike micropolar continua where it contributes to elastic energy, the wryness tensor appears only in the fracture energy in our approach. Thus, without damage, the solid conforms to the classical continuum. By making suitable modifications to the terms within the elastic energy and applying the principle of virtual work, we arrive at the governing partial differential equations (PDEs). For assessing the proposed framework, we choose a specific form of energy density and demonstrate, through numerical examples, the effect of the newly introduced parameters. The classical phase-field model is readily recovered by switching off the micro-rotation. Additionally, we explore a potential application of this model in representing and propagating initial defects.

Examining a Conservative Phase-Field Lattice Boltzmann Model for Two-Phase Flows

In this study, a conservative phase-field lattice Boltzmann model is examined for simulating immiscible two-phase flows with large density and viscosity ratios. This model is built upon the Allen–Cahn equation, incorporating a filtered collision operator and high-order corrections in the equilibrium distribution functions. The numerical model is also integrated with the volumetric boundary scheme and the immersed boundary method to achieve various stationary and moving boundary conditions on arbitrary geometries for different application scenarios. A comprehensive evaluation of this phase-field solver is conducted through a range of academic and industrial cases. First, droplet splashing cases at different Reynolds numbers are studied. The phase-field LB model effectively captures the surface wave motion and splashing of the droplet, although some smearing of small structures is observed due to the diffused interface property of the numerical model. Second, a tuned liquid damper is simulated, accurately capturing sloshing forces on solid walls. In the simulation of a dam-breaking case, predicted gauge pressure and free-surface elevation time histories compare well with experimental data. Lastly, a gearbox case under dip lubrication is simulated, with both the gearbox churning loss and oil distribution being well predicted across a range of rotating speeds. The validation studies demonstrate the effectiveness of the present phase-field lattice Boltzmann model in simulating immiscible two-phase flows with large and realistic density and viscosity ratios. The strengths, limitations, and weaknesses of the conservative phase-field LB model are discussed, and suggestions for the development and application of this numerical model are provided.

Theoretical Analysis of Self-Shrinkage Spheroidization of Irregular Degradable Polymer Powder under Thermodynamic Nonequilibrium State

Laser three-dimensional (3D) printing offers significant advantages in integrating the shape and function of regenerative tissues through biomimetic manufacturing. However, its effectiveness is limited by the lack of specialized biopolymer powders—while solvent methods that use residual solvents produce powders with poor biocompatibility, mechanical methods result in irregularly shaped crystals. In this study, a biopolymer powder spheroidization and shaping technology, which utilizes the evolution of irregular powders into spheres with minimal surface free energy in the molten state, is proposed based on the thermodynamic principle of minimum energy. Initially, the motion trajectory and temperature field of the poly(L-lactic acid) (PLLA) powder during spheroidization were quantitatively assessed and optimized using Stokes’ law and Fourier's principle. Subsequently, the cohesive forces and aggregation kinetics of the polymer chains were calculated using molecular dynamics. Finally, based on these calculations, a phase-field model was constructed to simulate the evolution of the spheroidization rate and deduce the optimal parameters for the process. This precise approach enhances PLLA spheroidization control for laser 3D printing, improves part densification and surface quality, and offers a clean and efficient path for preparing high-quality PLLA spheroidized powder for laser 3D printing.

Fatigue crack growth in functionally graded materials using an adaptive phase field method with cycle jump scheme

Functionally graded materials provide versatility in adjusting the volume fractions of constituent materials to meet specific design requirements. However, this customization often introduces mode-mixity at the crack tip, posing challenges in predicting fracture under cyclic loading with discrete approaches and computationally expensive with conventional phase-field fracture models. To address these issues, this paper introduces an adaptive phase-field fracture formulation with cycle jump scheme to elegantly predict fatigue crack nucleation and growth in functionally graded materials. Within this framework, the effective properties at a point are estimated using the Mori–Tanaka homogenization scheme, while the crack growth due to cyclic load is captured by incorporating an additional fatigue degradation parameter. Moreover, the computational efficiency of the proposed framework is improved through an adaptive mesh refinement and explicit cycle jump scheme. The adaptive refinement scheme utilizes an error indicator derived from both the displacement solution and phase-field variable. The adaptive refinement scheme is integrated with efficient quadtree decomposition, which generates a hierarchical mesh structure. Hanging nodes resulting from the quadtree decomposition are efficiently handled using a polygonal finite element method. The proposed framework is validated against experimental and numerical results reported in the literature. Furthermore, we investigate the fatigue crack growth resistance across a broad range of material gradation directions, gaining valuable insights and identifying functionally graded materials with high fatigue resistance.

Unconditional Energy Stable Runge-Kutta Schemes for a Phase Field Model for Diblock Copolymers

Unconditional Energy Stable Runge-Kutta Schemes for a Phase Field Model for Diblock Copolymers

Effect of porosity gradient on fracture mechanics of bi-directional FGM structures: Phase field approach

This research aims to develop a computational model that can accurately predict the fracture behavior of porous bi-directional Functionally Graded Materials (FGMs). The Voigt model for homogenization, is established to account the effects of porosity fraction and gradient distribution within the FGMs, providing valuable insights about the brittle crack propagation. The study employs the UMAT subroutine in ABAQUS software and establishes an analogy between the phase field evolution law and the heat transfer equation, enabling efficient analysis of complex fracture problems. To validate the model, 2D fracture benchmark cases are analyzed, demonstrating its ability to capture different failure modes and the intricate material behavior of porous FGMs under fracture conditions. Furthermore, newly parametric analyses, that highlights the impact of various values of porosity’s volume fraction and FGM’s power law indexes on the brittle fracture path, are conducted to further validate the effectiveness of the newly developed phase field model in predicting the fracture behavior of bi-directional porous FGMs.

Hydrodynamics of multicomponent vesicles: A phase-field approach

In this paper, we introduce a thermodynamically-consistent phase-field model to investigate the hydrodynamics of inextensible multicomponent vesicles in various fluid flows with inertial forces. Our model couples the fluid field, surface concentration field representing chemical species on the membrane, and vesicle dynamics, while enforcing global area and volume constraints through a Lagrange multiplier method. Specifically, we employ full Navier–Stokes equations for the fluid field, the Cahn–Hilliard equations for the species concentration on the membrane, a nonlinear advection–diffusion equation to describe the evolution of the vesicle membrane, and an additional equation to enforce local inextensibility. We utilize a residual-based variational multiscale method for the Navier–Stokes equations and a standard Galerkin finite element framework for the remaining equations. The PDEs are solved using an implicit, monolithic scheme based on the generalized-α time integration method. We extend previous models for homogeneous vesicles (Aland et al., 2014; Valizadeh and Rabczuk, 2022) to multicomponent vesicles, introducing Cahn–Hilliard equations while maintaining thermodynamic consistency. Additionally, we employ isogeometric analysis (IGA) for higher accuracy. We present a variety of two-dimensional numerical examples, including multicomponent vesicles in shear flow and Poiseuille flow with and without obstructions, using the resistive immersed surface method to handle obstructions. Furthermore, we provide three-dimensional simulations of multicomponent vesicles in Poiseuille flow.

Mechanics of Materials: Multiscale Design of Advanced Materials and Structures

Materials can now be designed and architectured like structural components for targeted mechanical and physical properties. Structures and microstructures should not be studied independently and their design will benefit from a multiscale approach combining nonlinear continuum mechanics approaches and physical descriptions of elasticity, viscoplasticity, phase transformations and damage of microstructures, at various scales. The aim of the workshop was to gather outstanding junior and senior researchers in the various branches of mathematics, physics and engineering sciences suited to address the question of design of materials and structures by means of multiscale discrete and continuum approaches to their constitutive behavior. Examples include atomic or macroscopic lattices, random or periodic cellular materials, smart materials like shape memory alloys, 3D woven composites, acoustic and electromagnetic metamaterials, etc. Modern continuum mechanics relies on sophisticated constitutive laws for anisotropic materials exhibiting elastoviscoplastic behavior, still a field of intense research with new mathematical concepts. In particular size-dependent properties are addressed by resorting to generalized continua such as gradient or micromorphic and phase field models. The latter are attractive for the simulation of microstructure evolution coupled with mechanics, due to thermodynamic and metallurgical processes and damage. Scale transition and homogenization methods for continuous and discrete systems are required for the determination of effective material and structural behavior. Metamaterials are architectured materials specifically designed to achieve certain propagation and dispersion properties of elastic and plastic waves. Optimization strategies for the design of optimal architectures are involved in the design process. Target functions for optimization are now based on multicriteria (stiffness, strength, thermal expansion, transport properties, anisotropy etc.).

Phase-field modeling of peritectic coupled growth in the TRIS–NPG model system

Microstructures forming during the directional solidification of the hypo-peritectic TRIS–NPG model alloy were studied by phase-field simulations. Steady state growth forms could be obtained under conditions similar to those in recent space experiments. In two dimensions, the wavelength range of stable lamellar structures has been determined as function of the temperature gradient and the melt composition. Front temperatures extracted from simulations were compared to the predictions of the modified Jackson–Hunt theory. In three dimensions, structures and phenomena known from eutectic systems were reproduced. Transitions between lamellar and rod structures were induced by slowly changing the melt composition. The regularizing effect of tilting the temperature gradient, transforming a random labyrinth pattern to an ordered one, has also been demonstrated. These simulations show the universality of the basic governing mechanism — that is, the competition of solute diffusion with capillary effects — in the pattern formation of these multi-phase systems.