Phase Field Research Articles

Phase field fracture in elastoplastic solids: a stress-state, strain-rate, and orientation dependent model in explicit dynamics and its applications to additively manufactured metals

Phase field models have gained growing popularity in analysing fracture behaviour of materials. However, few have been explored to simulate dynamic ductile fracture to date. This study aims to develop a phase field framework considering strain rate, stress state, and orientation dependent ductile fracture under dynamic loading. Firstly, governing equations of displacement and phase fields are formulated in an explicit finite element framework. Secondly, constitutive relations are established with a hypoelastic-plasticity framework, encompassing the influence of material orientation and strain rate on both plasticity and fracture initiation. Stress states dependent fracture initiation is also considered. Thirdly, finite element implementation and corotational formulation for constitutive equations are derived. To validate the proposed model, finally, additively manufactured samples, involving material-level and crack propagation specimens are tested under dynamic loading conditions. Overall, the proposed phase field model can properly reproduce the experimental force-displacement curves and crack paths. Uniaxial tension tests reveal that a higher strain rate can lead to a higher hardening curve and lower ductility. Other material specimens further demonstrate the capability to predict stress state and orientation dependent dynamic fracture. To simulate dynamic crack paths accurately, it is necessary to consider anisotropic fracture initiation. Lastly, the phase field model was applied for the first time to predict the dynamic response of triply periodic minimal surface (TPMS) structures. Dynamic crack patterns were effectively captured, and the fracture mechanisms were thoroughly analysed. This study provides an explicit phase field framework for dynamic ductile fracture, with applications to additively manufactured materials and structures.

Response of the Indian Ocean Meridional Overturning Circulation to the Subtropical Indian Ocean Dipole

Abstract Using the German contribution to Estimating the Circulation and Climate of the Ocean, version 3 (GECCO3), reanalysis data, this work explores the Indian Ocean meridional overturning circulation (IMOC) variability and mechanism during the mature phase of the subtropical Indian Ocean dipole (SIOD). The IMOC is decomposed into the Ekman component, geostrophic component, external mode, and residue. The IMOC exhibits counterclockwise circulation anomaly in 0°–30°S during the mature phase of the SIOD. While the Ekman component dominates in 10°–30°S, the geostrophic component prevails in 5°–20°S. During the mature phase of the positive SIOD events, while an anticyclonic wind anomaly over the southern Indian Ocean causes a convergence and sinking of the seawater near 30°S, a cyclonic wind field anomaly near 10°S induces a divergence and rising, causing a counterclockwise Ekman component anomaly in 10°–30°S. The geostrophic component anomaly in 5°–20°S is caused by the sea level anomaly (SLA) gradient around 10°S related to a Rossby wave. A linear, 1D baroclinic Rossby wave model shows that the negative SLA in the west is caused by local and remote wind forcing, whereas the positive SLA in the east is generated by radiation from the eastern boundary and is slightly contributed by local wind forcing. Further, Parallel Ocean Program, version 2 (POP2), experiments confirmed that the Ekman component anomaly primarily responds to the wind field of the mature phase of the SIOD and revealed that the geostrophic component anomaly is affected by the wind field of both the developing and mature phases of the SIOD.

Combination of high-throughput phase field modeling and machine learning to study the performance evolution during lithium battery cycling

Combination of high-throughput phase field modeling and machine learning to study the performance evolution during lithium battery cycling

Enhanced cyclic stability of NiTi shape memory alloy elastocaloric materials with Ni4Ti3 nanoprecipitates: Experiment and phase field modeling

Enhanced cyclic stability of NiTi shape memory alloy elastocaloric materials with Ni4Ti3 nanoprecipitates: Experiment and phase field modeling

An FFT based chemo-mechanical framework with fracture: Application to mesoscopic electrode degradation

An FFT based method is proposed to simulate chemo-mechanical problems at the microscale including fracture, specially suited to predict crack formation during the intercalation process in batteries. The method involves three fields fully coupled, concentration, deformation gradient and damage. The mechanical problem is set in a finite strain framework and solved using Fourier Galerkin for non-linear problems in finite strains. The damage is modeled with Phase Field Fracture using a stress driving force. This problem is solved in Fourier space using conjugate gradient with an ad-hoc preconditioner. The chemical problem is modeled with the second Fick’s law and physically based chemical potentials, is integrated using backward Euler and is solved by Newton–Raphson combined with a conjugate gradient solver. Buffer layers are introduced to break the periodicity and emulate Neumann boundary conditions for incoming mass flux. The framework is validated against Finite Elements the results of both methods are very close in all the cases. Finally, the framework is used to simulate the fracture of active particles of graphite during ion intercalation. The method is able to solve large problems at a reduced computational cost and reproduces the shape of the cracks observed in real particles.

Phase separation behavior of polymer modified asphalt by molecular dynamics and phase field method: A review

Phase separation behavior of polymer modified asphalt by molecular dynamics and phase field method: A review

Investigation of oxygen transport in porous transport layer with different porosity gradient configurations using phase field method

Minimizing oxygen accumulation in the porous transport layer (PTL) is crucial for reducing mass transfer losses in proton exchange membrane (PEM) electrolyzer. This study develops a two-dimensional transient model of gas-liquid two-phase flow at the anode of PEM electrolyzer using the phase field method. The model investigates the mechanisms of oxygen transport and the interactions among various oxygen paths in PEM electrolyzer. We explore the impact of porosity gradient configurations in the PTL and the presence of a surface microporous layer (MPL) on oxygen transport. The findings indicate that for PTL with an average porosity of 60%, forward gradient configuration—where porosity increases from the catalyst layer (CL) towards the channel (CH)—promotes the merging of bubble sites and path contraction, thereby reducing oxygen saturation. The optimal gradient configuration, with porosities of 50% at the CL and 70% at the CH, achieves a 29.5% reduction in oxygen saturation. Conversely, reverse gradient configuration, with decreasing porosity from CL to CH, results in increased oxygen saturation. The addition of surface MPL further lowers oxygen saturation and shortens oxygen breakthrough time; smaller MPL particle sizes correspond to lower oxygen saturation and shorter breakthrough times. This study provides valuable insights for the optimal design of PTL structures in PEM electrolyzers.

Analysis of CO2 bubble growth detachment kinetics in direct methanol fuel cell flow channels

CO2 bubbles flow in the anode flow channel is an important issue in the commercialization process of direct methanol fuel cells (DMFC). A T-channel model is built in COMSOL using the phase field method to investigate the CO2 bubbles flow at the anode side of the DMFC. The factors and mechanisms of single bubble detachment are discussed by analyzing the methanol inlet velocity (Ul), CO2 inlet velocity (Ug), contact angle of the diffusion layer, and the Weber number during bubble detachment. The research findings indicate that increasing the liquid flow rate leads to the generation of smaller bubbles that detach more rapidly due to the increases of drag force (FD) and the shear-life force (FSL) to overcome the surface tension on the bubble. The CO2 inlet velocity can promote the bubble detachment due to the increase in FSL, but also leads to a larger detachment diameter. Compared to hydrophobic surfaces, hydrophilic surfaces are more conducive to bubble detachment and removal. In all case We (Weber number) is significantly less than 0.6, indicating that liquid momentum dominated the bubble detachment process. Once the ratio of the gas momentum to the liquid one is greater than 1, the bubble is hard to detach. The contour map of bubble flow patterns and the bubble detachment diameters distribute with the ratio of Ug/Ul can further indicate that the bubble detachment is connected with the ratio of Ug/Ul closely, which will have guiding significance for the selection of inlet velocity of DMFC.

Calculation method for brittle fracture of functional gradient materials

Combined the phase field model with the wavelet dummy node-virtual crack closure technique (WDN-VCCT) used for stress intensity factors (SIFs) calculation. Calculated the node displacement using an improved brittle fracture phase field method, established the relationship between node displacement and node force using WDN-VCCT, and calculated the SIFs at the crack tip. The correctness and accuracy of the proposed method were verified through functional gradient material (FGM) tensile experiment. The influence of crack inclination angle, crack position and gradient index on mechanical response, and SIFs values at the crack tip was discussed. This study provides important computational tools for estimating the service life of FGMs and important references for structural optimization design methods.

Phase field modelling of tunnel excavation damage in transversely isotropic rocks

Layered rocks, prevalent in geological formations, exhibit complex failure behaviours driven by pronounced anisotropy in their mechanical properties. However, the intricate failure mechanisms remain inadequately understood due to the spatial complexity of these mechanical responses. To address this challenge, this study proposes a novel phase field model tailored for layered rocks. A new strain energy decomposition method is introduced, uniquely formulated based on stress components and designed for transversely isotropic constitutive models. Numerical solutions to the coupled stress-phase field equations are obtained using user-defined element and material subroutines, implemented through a staggered solution scheme. The accuracy and robustness of the numerical approach are validated through simulations of a composite panel with a central hole under tensile loading. Additionally, the model is applied to the excavation of the layered soft rock tunnel, revealing the significant influence of bedding plane angles on displacement and stress field distributions, as well as post-excavation failure modes. Notably, the model captures the predominant shear-slip failure along bedding planes following excavation, effectively reflecting the complex mechanical interactions in layered rocks. This proposed model represents a significant advancement in understanding the failure mechanisms of layered rocks, providing a valuable tool for future geotechnical applications.

Modeling abscission of cacti branches

During evolution, various functional principles have evolved that allow plants to create predetermined breaking points for the spatially defined abscission of organs. In the plant family of cacti, some species, such as Cylindropuntia bigelovii, have fragile branch–branch junctions that serve vegetative reproduction, while in other species, such as Opuntia ficus-indica, they are very stable. The fracture behavior of these junctions has been thoroughly characterized anatomically and mechanically, the data being the prerequisite for the performance of cactus-inspired phase field simulations. We have found that models composed of homogeneous materials or material systems with low elastic modulus contrast (analogous to Cylindropuntia bigelovii) exhibit a fracture mode where cracks initiation occurs at the epidermis of the junction notch. In comparison, heterogeneous material systems with high elastic modulus contrast (similar to Opuntia ficus-indica) show fracture nucleation along the inner vascular bundles, with an increase in the maximum fracture energy by a factor of 2.2. In the high contrast heterogeneous models, the V-notch and stiffening of the dermal tissue (“periderm”) have a negligible effect on their fracture behavior. In addition, the fracture morphologies of these models resemble the rough junction fracture sites found experimentally. The knowledge gained about the geometric influences and the importance of the contrasts in the mechanical properties of the individual materials in the overall system can be transferred as functional principles to bioinspired engineering composites in order to program their fracture behavior.

Assessing grain boundary variability through phase field crystal simulations

Characterization of grain boundaries (GBs) typically focuses exclusively on ideal minimum-energy structures, thus offering a limited perspective on the potential structural diversity and related property variations of GBs. In this study, phase field crystal (PFC) simulations are employed to systematically explore alternative GB states through γ-surface sampling. A large number of tilt and twist GBs are examined in both fcc and bcc bicrystal structures. It is demonstrated that identifying variants in GB structure necessitates considering a set of microscopic degrees of freedom (DOF), comprising the components of relative crystal translation, in addition to the GB's five macroscopic DOF. Taking GB energy and excess volume as examples of key GB properties, a significant spread in GB energy is revealed, stemming from variations in the microscopic DOF, while maintaining constant macroscopic DOF. In addition, the significant variations found in GB excess volume and energy when GB variants are considered challenge the common assumption of a strong correlation between them. Taken together, the findings underscore the importance of recognizing a range of GB structures and properties for each macroscopic GB configuration, rather than relying on singular ideal minimum-energy GB structures, as is usually done. Published by the American Physical Society 2024

Setting the standard for machine learning in phase field prediction: a benchmark dataset and baseline metrics

Phase field models are an important mesoscale method that serves as a bridge between the atomic scale and the macroscale, used for modeling complex phenomena at the microstructure level. Machine learning can be employed to accelerate these simulations, enabling faster and more efficient analyses. However, the development of new machine learning algorithms depends on access to extensive datasets. This work introduces an accessible and well-documented dataset aimed at benchmarking new machine learning algorithms. We validate the dataset with a benchmark using U-Net regression, a widely used neural network architecture. Although direct comparisons are limited by the lack of existing benchmarks, our model’s error metrics are competitive with previous work and generalize across multiple domain sizes. This contribution provides a valuable resource for future efforts in machine learning model development for phase field simulations and demonstrates the potential of U-Net regression, highlighting the scope for novel method development in this area.

Diffusive-length-scale adjustable phase field fracture model for large/small structures

In phase field fracture (PFF) method, the sharp crack is approximated by a phase field crack zone whose size is characterized by a diffusive length scale. Recently, the diffusive length scale is usually regarded as a constant material parameter determined by the fracture toughness, material strength, and young's modulus. As a result, the application of the PFF method poses challenges when dealing with structures whose sizes are much too large or small compared to the constant diffusive length scale. In details, for a large-scale structure, a significant computational burden is inevitable due to the limitation imposed by the constant diffusive crack length scale on the element size (i.e. the element size is generally at least smaller than half of the diffusive crack length scale to achieve the sufficient precision for the phase field process zone). For a small-scale structure, the crack patterns tend to be unclear and unrealistic due to the excessively large diffusive crack zone. To address these limitations, we propose a novel PFF method to make the relation between the diffusive length scale and the material parameters adjustable via modifying the energetic degradation functions. It is found that with the increase of the diffusive crack length scale, a transition from quasi-brittleness to brittleness is observed in the constitutive relationship curves, which coincides with the classical size effect on structural strength. Further, the Bažant's size effect in classical fracture mechanics can be reproduced by the present PFF method through scaling the size of a geometrically similar structure, i.e. a large-scale structure exhibits toughness-dominated fracture while a small-scale structure behavior strength-dominated fracture. The present PFF method efficiently addresses the mismatch of diffusive length scales for various material constituents by examining phase field crack growth in particle-reinforced composite plates. By adjusting the diffusive crack length scale based on structure size, it avoids high computational costs from large structures and unrealistic crack patterns from small ones. Moreover, it outperforms traditional PFF methods using a constant diffusive length scale in handling composite materials.

Moving Next-Generation Phase-Field Models to BMS Applications - a Case Study That Confirms Professor Uzi Landau's Foresight

Physics-based electrochemical models play an important role in the model-based analysis, virtual engineering, estimation, control, and Battery Management Systems (BMS) of lithium-ion batteries.1 The diffusion process within the cell electrodes has typically been modeled using Fick's laws of diffusion.2 Recently, the intercalation of lithium ions in the electrode particle was posed as a phase change problem, where the area intercalated with lithium undergoes a phase change compared to the pristine material. Researchers have approached this phase change problem in terms of evolving core-shell problems3,4 and, more recently, using phase field models5,6 that may hold certain advantages over the traditional Fickian diffusion models. In this paper, we briefly discuss the relevance and rich physics of phase-field models and show how proper numerical analysis and schemes can help with implementing these models in BMS applications. Further, an investigation of phase-field model-based optimization framework predicts an impulse-like control profile and was designed to reduce capacity degradation. This work was partially inspired by the pulse-charging protocol proposed by Professor Landau in his 2006 work7. An open-source framework is developed and will be shared for predicting the (im)pulse protocol reported in this.

Efficient Reformulation of Lithium-Ion Battery Models Using Symmetric Polynomials: Extending to Phase Field Models

Energy storage will play a crucial role in the transition to clean and renewable sources of energy production to meet future consumption demands. Lithium-ion (Li-ion) batteries are widely adopted due to their high energy density, low self-discharge rate and fast charging capabilities for applications like electric vehicles and grid storage.Lithium-ion batteries are typically modeled using porous electrode theory coupled with various transport and reaction mechanisms, along with suitable discretization or approximations for the solid-phase diffusion equation.The solid-phase diffusion equation represents the main computational burden for typical pseudo-2-dimensional (p2D) models since these equations in the pseudo r-dimension must be solved at each point in the computational grid. This substantially increases the complexity of the model as well as the computational time. Traditional approaches towards simplifying solid-phase diffusion possess certain significant limitations, especially in modeling emerging electrode materials which involve phase changes and variable diffusivities. A computationally efficient representation for solid-phase diffusion was shown in a previous paper1 based on symmetric polynomials using Orthogonal Collocation and Galerkin formulation (weak form). This approach increases the accuracy of the approximation (p form in finite element methods) to enable efficient simulation with a minimal number of semi-discretized equations, ensuring mass conservation even for non-linear diffusion problems involving variable diffusivities.These methods will be reviewed for the full p2D model, illustrating their advantages in simulating high C-rates and short-time dynamic operation and extended to newer systems containing phase field models to analyze the robustness of the approximations. References R.S. Thiagarajan, A. Subramaniam, S. Kolluri, T. R. Garrick, Y. Preger, 729 V. De Angelis, J.-h. Lim, V. R. Subramanian et al2023 J. Electrochem. Soc. 170 010528

The reverse-encapsulation and core-release process of a compound droplet under thermocapillary effects

This work demonstrates that a core-shell compound droplet can undergo separation into two different phase droplets or be transformed into a reverse-encapsulation droplet under the influence of thermocapillary effects. This study comprehensively investigates the thermocapillary effects on the migration, reverse-encapsulation, and core-release dynamics of compound droplets, with particular emphasis on the interactions among key dimensionless parameters, including the Marangoni number, Reynolds number, and the radius ratio of the inner to the middle fluid of a core-shell compound droplet. Using a reduction-consistent phase field model for incompressible N-phase flows, this research provides an in-depth analysis of the reverse-encapsulation and core-release processes of a compound droplet. The reverse-encapsulation process is characterized by three distinct stages: The initial acceleration stage, the encapsulation stage, and the post-transformation stage. Similarly, the core-release process also has three stages: The initial acceleration stage, the middle fluid contraction stage, and the independent movement stage. The results indicate that the Marangoni number has little influence on the migration velocity, while the tangential surface force (thermocapillary force) is the cause of these processes. The Reynolds number has a significant effect on the encapsulation dynamics, with higher values leading to slower and more turbulent processes. In addition, the radius ratio plays a critical role in determining the reverse-encapsulation and core-release processes, as well as migration behaviors. These findings advance our understanding of multiphase fluid dynamics and have significant implications for applications in microfluidics, targeted drug delivery, and advanced materials synthesis.

Three-dimensional fracture of UO2 ceramic pellets by phase field modeling

Three-dimensional fracture of UO2 ceramic pellets by phase field modeling

Stochastic analysis of dynamic fracture of concrete using CT-image based mesoscale models with a rate-dependent phase field method

Concrete structures are commonly exposed to dynamic loads spanning a wide range of strain rates, and the inherent mesoscale heterogeneities complicate stochastic dynamic fracture mechanisms even more. This work develops a numerical framework using mesoscale concrete models based on micro computed tomography (CT) images to investigate such mechanisms with meaningful stochastic analyses. A rate-dependent phase field model is proposed to characterise the dynamic initiation and propagation of cracks by incorporating both micro-viscosity and macroscopic viscoelasticity, which is described by two standard Maxwell elements with different relaxation times to consider a wide range of strain rates. Moreover, the viscoelastic constitutive relation is formulated in the full strain space, which allows for a spectral decomposition of the strain tensor to determine the effective damage driving force, thus effectively addressing the issue of compressive fracture. A numerical implementation scheme is developed by combining user-defined element and material subroutines in ABAQUS/Explicit solver. Extensive Monte Carlo simulations of dynamic tension up to a strain rate of 200 s−1 are performed with statistical analyses. This work reveals the intricate dynamics associated with mesoscale heterogeneities and identifies the critical transition state at 20 s−1. The transition is characterised by changing modes of fracture patterns, stress wave propagation, and load-carrying capacities. A new TDIF–strain rate–standard deviation relation is also proposed and aligns well with the increasing dispersion of experimental data. The relationship between void content and tensile strength reflects the formation characteristics of crack networks, with the void content exhibiting a positive correlation with the TDIF from 20 s−1 to 100 s−1.

An adaptive mesh refinement algorithm for crack propagation with an enhanced thermal–mechanical local damage model

This paper presents a computationally effective approach for crack propagation under mechanical and thermal loads based on an adaptive mesh refinement (AMR) approach tailored for our recently developed enhanced local damage model. The mesh-dependent issue encountered in the classical local theories is effectively mitigated by incorporation of fracture energy and element characteristic length into the damage evolution function. Our previous research has demonstrated that being equipped by a novel equivalent strain derived from the bi-energy norm concept and a new damage criterion recently proposed by Mazars et al., the model provides results comparable to the reference experimental data as well as other numerical models based on non-local/gradient damage and phase field method. In the framework of computational efficiency using finite elements, we significantly enhance the performance of our enhanced local model by considering adaptive mesh refinement (AMR). The finite element mesh is locally refined in the damaged zone, and the mesh refinement is conducted on-the-fly during the analysis. For that purpose, the damage parameter whose information is stored at integration points is selected as an indicator to mark whether an element should be refined or not after every loading step. For quadrilateral element mesh, a quad-tree technique is utilized, meaning that each marked element is further divided into four smaller quadrilateral elements. The so-called hanging nodes appear during the process, and the elements are thus treated as n-gons and are constructed by the Laplace shape functions, instead of the usual Lagranges shape functions. To show the accuracy and effectiveness of the proposed scheme, several numerical examples involving homogeneous and heterogeneous materials are studied. In these examples, the damage is induced either by only mechanical loads or by both mechanical and thermal loads.