Phase-field Model Research Articles

Phase-field approach for fracture prediction of brittle cracked components

The versatility of the phase-field method (PFM) in predicting elusive fracture trajectories has attracted researchers in recent years; however, limited focus has been directed toward fracture load prediction for brittle materials. This study presents a computationally low-cost and straightforward framework based on phase-field modeling to predict the fracture loads and fracture initiantion angles of brittle cracked components under in-plane loading conditions. The framework was assessed through a comprehensive comparison between numerical and experimental data for various specimens and test configurations. Despite using 1D analytical formulations for calculating phase-field parameters, great accordance between empirical data and simulation results demonstrated the effectiveness of the proposed model. A technique that restricts the computation of the phase-field variable merely around the crack tip was applied to significantly reduce the computational costs of PFM simulations. Furthermore, we examined different numerical aspects and confirmed the robustness of the proposed procedure.

Multi-phase-field modeling of the dissolution behavior of stoichiometric particles on experimentally relevant length scales

The dissolution of stoichiometric particles within a melt plays a crucial role in various material processes. This study presents a comprehensive phase-field model to analyze the dissolution behavior of these stoichiometric particles under experimental conditions. Our approach addresses the classical phase-field challenges related to modeling stoichiometric compounds and scaling to experimentally relevant lengths in a multi-phase, multi-component context. To overcome the difficulties posed by stoichiometric compounds, we rederive the classical phase-field evolution equations for a multi-phase system, adopting a composition-independent free energy expression for the stoichiometric compound. Additionally, we extend Feyen’s high driving force model [Feyen and Moelans, Acta Materialia, 256 (2023)] to multi-component systems, allowing us to perform quantitative simulations for technologically relevant material systems at experimental length scales within a reasonable computing time. The model’s precision in capturing diffusion-controlled transformations, including dissolution, growth, and the Gibbs–Thomson effect, is validated against analytical solutions for a hypothetical system. The quantitative nature of the model is validated by applying it to the dissolution of Al2O3 particles in CaO–Al2O3–SiO2 slags. We break new ground by conducting three-dimensional simulations for a system size of 875μm×875μm×875μm, directly comparable to confocal scanning laser microscopy experiments, where previous models were limited to two-dimensional simulations and a system size of 2μm×2μm. This validation underscores the model’s proficiency to quantitatively describe the diffusion-controlled dissolution of Al2O3 at the experimentally relevant length scales.

Multiscale modeling of diffuse damage and localized cracking in quasi-brittle materials under compression with a quadratic friction law

The diffuse damage and localized cracking of quasi-brittle materials (i.e., rocks and concretes) under compression can be delineated by a matrix-microcrack system, wherein a solid matrix phase is weakened by a large number of randomly oriented and distributed microcracks, and the macroscopic cracking is formed by a progressive evolution of microcracks. Several homogenization-based multiscale models have been proposed to describe this matrix-microcrack system, but most of them are based on a linear friction law on the microcrack surface, rendering a linear strength criterion. In this paper, we propose a new quadratic friction law within the local multiscale friction-damage (LMFD) model to capture the plastic distortion due to frictional sliding along the rough microcrack surface. Following that, a macroscopic Ottosen-type nonlinear strength criterion is rationally derived with up-scaling friction-damage coupling analysis. An enhanced semi-implicit return mapping (ESRM) algorithm with a substepping scheme is then developed to integrate the complex nonlinear constitutive model. The performance of LMFD model is evaluated compared to a wide range of experimental data on plain concretes, and the robustness of ESRM algorithm is assessed through a series of numerical tests. Subsequently, to effectively describe the localized cracking process, a regularization scheme is proposed by combining the phase-field model with the established LMFD model, and the discretization independent crack localization is numerically verified.

Phase field modeling of crack propagation in structures under tensile stress

Phase field modeling of crack propagation in structures under tensile stress

Tutorials: Physics-informed machine learning methods of computing 1D phase-field models

Phase-field models are widely used to describe phase transitions and interface evolution in various scientific disciplines. In this Tutorial, we present two neural network methods for solving them. The first method is based on physics-informed neural networks (PINNs), which enforce the governing equations and boundary/initial conditions in the loss function. The second method is based on deep operator neural networks (DeepONets), which treat the neural network as an operator that maps the current state of the field variable to the next state. Both methods are demonstrated with the Allen–Cahn equation in one dimension, and the results are compared with the ground truth. This Tutorial also discusses the advantages and limitations of each method, as well as the potential extensions and improvements.

Phase-field investigation of dendrite suppression strategies for all-solid-state lithium metal batteries

All-solid-state Li metal batteries are widely considered as the most promising technologies to realize the increasing safety and capacity requirements for the next generation of Li batteries. However, it is experimentally demonstrated that the penetration of stiff solid electrolytes by Li dendrites seems even more severe than that of liquid electrolytes and the abnormal failure behavior of soft Li metal penetrating hard SEs is quite elusive. In the present work, a multi-coupling phase field model of Li electrodeposition is further developed to incorporate the elasto-plastic mechanical behavior and the significant size effect of Li metal. Then, the established approach is utilized to specifically evaluate the different mechanisms of Li electrodeposition and stripping in liquid electrolytes and hard solid electrolytes with various governing factors including applied voltage, external stress, ionic conductivity, SEI inhomogeneity and interfacial porosity. Especially, we summarize the accurate quantification of the regularity of external stress mechanisms which determine whether the external stress can either promote or demote the nonuniformity of Li/ solid electrolytes interface. This research provides a theoretical perspective of mechanical mechanism to reveal the conflict of experimental optimization strategies and may be instructive for the Li/ solid electrolytes interface design of all-solid-state Li metal batteries.

Fast implicit update schemes for Cahn–Hilliard-type gradient flow in the context of Fourier-spectral methods

This work discusses a way of allowing fast implicit update schemes for the temporal discretization of phase-field models for gradient flow problems that employ Fourier-spectral methods for their spatial discretization. Through the repeated application of the Sherman–Morrison formula we provide a rule for approximations of the inverted tangent matrix of the corresponding Newton–Raphson method with a selectable order. Since the representation of this inversion is exact for a sufficiently high approximation order, the proposed scheme is shown to provide a fixed-point-type iterative solver for gradient flow problems that require the solution of linear systems in the context of an implicit time-integration. While the proposed scheme is applicable to general gradient flow phase-field models, we discuss the scheme in the context of the Cahn–Hilliard equation, the Swift–Hohenberg equation, and the phase-field crystal equation for which we demonstrate the performance of the proposed method in comparison with classical solvers.

A unified FFT framework with coupled gradient damage model and phase-field method for the failure analysis of composites

A unified FFT framework coupling the Gradient Damage Model (GDM) and the Variational Phase-Field Model (PFM) for failure analysis of composites is proposed. GDM and PFM are used to simulate the failure of the matrix and the interphase, respectively. Based on the similarity of the damage evolution equations (elliptic partial differential equations) of GDM and PFM, a unified iterative solution and numerical implementation based on fast Fourier transform (FFT) are derived, and simultaneously applied to a computational model. In addition, the high computational efficiency of the FFT method is used to solve the mechanical equilibrium problem. The complete form of the PFM is introduced to avoid erroneous damage diffusion at the different phase interfaces of the composites, and a stress decomposition-based PFM is formulated. 2D and 3D computational examples are established to investigate the mesh insensitivity of the damage models used and the erroneous damage diffusion at interfaces. The applicability of the stress decomposition-based PFM at interphase is discussed in the context of single-edge notched specimens under tension and fiber-reinforced composites with interfacial phases under transverse tension. Finally, the performance of the unified computational framework is evaluated using two simple numerical examples, and the mechanical behavior of unidirectional CF/PEEK composites under transverse tensile loading is simulated based on the unified framework. The results are compared with experiments to evaluate the computational accuracy and practical applicability of the proposed framework.

Enhancing plasticity in BCC Mg-Li-Al alloys through controlled precipitation at grain boundaries

This study investigates the improvement of plasticity in body-centered cubic magnesium (Mg)-lithium (Li)-aluminum (Al) alloys, crucial for lightweight structural applications. The ternary Mg-Li-Al alloys exhibits high strength but low ductility. Precipitates at grain boundaries in these alloys, linked to reduced plasticity, are examined for their crystal structure and composition. Advanced microscopic techniques reveal the transformation of precipitates and the development of specific structures at grain boundaries. Thermodynamics of element diffusion at grain boundaries are explored through first-principles calculations, and a phase-field simulation models precipitate evolution. Molecular dynamics simulations elucidate nanoscale mechanisms governing the transition from brittle to ductile fracture modes during artificial aging. The D03−Mg3Al at grain boundaries is a brittle phase, and through a 170 °C aging treatment, it induces the precipitation of lamellar α-Mg phase with D03−Mg3Al as nucleation sites. The occupancy energy of Al atoms at Li sites in α-Mg is found to be lower than that in D03−Mg3Al, leading to the dissolution of D03−Mg3Al. The α-Mg, characterized by a stronger metallic nature, exhibits a better-matched modulus with the matrix and enhanced dislocation mobility. The precipitation of α-Mg plays a pivotal role in significantly improving the ductility of the alloy. This work contributes to the understanding of the complex interplay between alloy composition, grain boundary precipitates, and plasticity, as well as brings insights to guide interfacial control in the development of advanced Mg-Li-Al alloys for structural applications.

Thermodynamically consistent phase field model for liquid-gas phase transition with soluble surfactant

Despite the enormous potential in facilitating natural development and migration of interfaces during multiphase simulation, the phase-field method remains restricted to low-density ratios, owing to inherent thermodynamic inconsistency, especially for multiphase flow systems with surfactants. The present paper first constructs a liquid-vapor phase transition phase-field model with soluble surfactants using the second law of thermodynamics as the original model. Then, a simplified liquid-vapor phase transition model with soluble surfactants that satisfies thermodynamic consistency is proposed to simulate pool boiling at higher-density ratio. A novel numerical algorithm for the simplified model that satisfies semi-discrete thermodynamic consistency is also developed. Compared with the original model, the thermodynamically consistent characteristics of the simplified numerical model proposed in this paper can significantly reduce the spurious velocity on the interface of a static droplet and thus enable the numerical model to simulate liquid-vapor transition at higher liquid/vapor density ratios. Vapor-liquid coexistence, Laplace's law, and multiple bubble coalescence are used to validate the accuracies and effectiveness of the mathematical model and numerical algorithm. The liquid/vapor density ratio can reach 6776:1 under saturation temperature 0.3Tc (Tc is the critical temperature). The approach is then used to model pool boiling at a low saturation temperature (0.5Tc) with and without soluble surfactants, significantly lower than reported in comparable literature. The results demonstrate that surfactants significantly influence the dynamics of bubbles, and a critical concentration can be identified. In addition, soluble surfactants can also suppress coalescence between adjacent bubbles and prevent the formation of larger bubbles during pool boiling.

Phase field modeling of fracture mechanics for ion-exchanged glass considering large viscoelastic deformation, mechanic–electrochemical coupling

A phase field model is proposed to investigate the fracture mechanics of ion-exchanged glass, which involves significant viscoelastic deformation and stress-diffusion interaction. Based on this theoretical model, the fracture behavior of both double-sided and single-sided ion exchange processes was studied. The results indicate that, compared to single-sided ion exchange, the double-sided process is more likely to induce crack initiation and propagation, ultimately resulting in a notable reduction in glass strength. For both cases, increasing the glass thickness and interfacial energy density will significantly improve the fracture resistance of the glass. Notably, when the glass thickness or interfacial energy density is large enough, the crack can be completely closed, ensuring the integrity of the glass during the ion exchange process. Furthermore, the study conducted a preliminary investigation of two industrially relevant conditions: edging and impurities. The impacts of edging parameters, glass interfacial energy density, and covering radius on glass stress evolution, mechanical deformation, crack initiation, and crack propagation during chemical strengthening were summarized. These findings offer valuable theoretical support and a reference for optimizing chemical strengthening process parameters and enhancing the strength of glass products.

A phase-field length scale insensitive mode-dependent fracture model for brittle failure

This article presents a consistent phase-field length scale insensitive mode-dependent fracture model for brittle failure by proposing mode-factor dependent degradation functions that incorporate the effect of two additional fracture parameters, namely the mode-II critical energy release rate and mode-II fracture strength, on the overall mechanical response. Using the proposed mode-factor dependent degradation functions and employing a recently proposed modified strain decomposition scheme, we provide analytical expressions for the mode-I and mode-II fracture strengths corresponding to the mode-dependent parts of the elastic energy. Adopting a modified Benzeggagh–Kenane (B–K) criteria, we propose mode-dependent driving forces by deriving expressions for the critical energy release rate corresponding to individual fracture modes. The proposed model provides a consistent coupling between the different fracture modes and can thus predict fracture for all possible mode-mixity ratios. A parametric study is carried out to unravel the effect of mixed-mode fracture parameters on the mechanical response of isotropic materials by considering a few representative numerical examples. The numerical results from the proposed model show an excellent agreement with experimental results reported in the literature.

Phase field modeling of underloads induced fatigue crack acceleration

This study proposes a novel methodology to model the fatigue crack growth acceleration under underloads using a phase field fracture framework. In this model, fatigue crack growth is characterized by the degradation of fracture toughness, with an emphasis on employing a representative loading strategy instead of explicit cyclic loading, thus accelerating simulations of high-cycle fatigue. The model integrates a zone-based crack acceleration approach that responds to single-cycle underload. Notably, the model adeptly captures the dynamics described by the Paris-Erdogan law. Post-underload crack growth acceleration is simulated by identifying an acceleration zone near the crack tip, inspired by existing models based on the plastic zone. This zone is defined by a strain energy density threshold, and the underload ratio governs the rate of fatigue damage accumulation attenuation within this area. The implementation leverages the UMAT user subroutine in Abaqus, utilizing coupled temperature-displacement elements where temperature analogously represents the phase field parameter. Experimental validation of the model confirms its ability to accurately reflect the loss of fatigue life and the acceleration of crack growth rates in compact tension and middle tension specimens. Additionally, the model shows promise for extension to periodic underloads, highlighting its potential for simulating real-world fatigue scenarios effectively.

Interface stabilization and propagation in phase field models of solidification: resolving the issue of large driving forces

Abstract One of the long-standing problems in the phase field, namely, combining the principles of thermodynamics and capillarity with the numerical aspects of interface propagation in simulations, is re-investigated. Numerical schemes are discussed which allow for stable simulations with arbitrary driving forces, considering or excluding capillarity. We re-investigate a classical stabilization scheme that decouples interface stabilization from curvature evaluation, ensuring stable simulations even under large driving forces. A novel mathematical analysis gives a rigorous estimate for the time stepping and a numerical value of the required stabilization strength. The proposed stabilization scheme is benchmarked for 3-dimensional dendritic growth under directional solidification conditions for different solidification speeds.

A thermodynamically consistent phase-field model and an entropy stable numerical method for simulating two-phase flows with thermocapillary effects

In this study, we have derived a thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects. This model accommodates variations in physical properties such as density, viscosity, heat capacity, and thermal conductivity between the two components. The model equations encompass a Cahn-Hilliard equation with the volume fraction as the phase variable, a Navier-Stokes equation, and a heat equation, and meanwhile maintains mass conservation, energy conservation, and entropy increase simultaneously. Given the highly coupled and nonlinear nature of the model equations, we developed a semi-decoupled, mass-preserving, and entropy-stable time-discrete numerical method. We conducted several numerical tests to validate both our model and numerical method. Additionally, we have investigated the merging process of two bubbles under non-isothermal conditions and compared the results with those under isothermal conditions. Our findings reveal that temperature gradients influence bubble morphology and lead to earlier merging. Moreover, we have observed that the merging of bubbles slows down with increasing heat Peclect number PeT when the initial temperature field increases linearly along the channel, while bubbles merge faster with heat Peclect number PeT when the initial temperature field decreases linearly along the channel.

A thermodynamic framework for ductile phase-field fracture and gradient-enhanced crystal plasticity

This study addresses ductile fracture of single grains in metals by modeling of the formation and propagation of transgranular cracks. A proposed model integrates gradient-extended hardening, phase-field modeling for fracture, and crystal plasticity. It is presented in a thermodynamical framework in large deformation kinematics and accounts for damage irreversibility. A micromorphic approach for variationally and thermodynamically consistent damage irreversibility is adopted. The main objective of this work is to analyze the capability of the proposed model to describe transgranular crack propagation. Further, the micromorphic approach for damage irreversibility is evaluated in the context of the presented ductile phase-field model. This is done by analyzing the impact of gradient-enhanced hardening considering micro-free and micro-hard boundary conditions, studying the effect of the micromorphic regularization parameter, evaluating the performance of the model in ratcheting loading and testing its capability to predict three-dimensional crack propagation. In order to solve the fully coupled global and local equation systems, a staggered solution scheme that extends to the local level is presented.

Voltammetric Study of Phase Separating Anode Materials in Li-Ion Battery Using Phase Field Model

The increasing demand for higher energy capacity in lithium-ion batteries has underscored the significance of modeling lithiation with alternative electrode materials, such as silicon (Si) and tin (Sn). This research addresses the need for comprehensive insights into the (dis)charging process, particularly focusing on the intricate phase transformations in the anode. Phase-field modeling has proven effective in capturing these transformations [2], revealing volumetric deformations of up to 300% during (dis)charging cycles [1]. Unfortunately, these deformations compromise material integrity, leading to reduced battery cycle life and reliability.To address these challenges, our work presents a novel framework for modeling lithiation in materials exhibiting two-phase diffusion using phase field modeling. Unlike previous models limited to single-phase diffusion, our framework provides a more realistic representation of the diffusion process. This realism is achieved by incorporating the time-dependent voltage applied to the anode, a crucial factor in understanding the electrochemical response during lithiation [3].Furthermore, we leverage the currents generated during the lithiation and delithiation processes to conduct a comprehensive voltammetric study. This study goes beyond existing research by considering the impact of phase separation on the currents developed during Li-ion battery operation. Our findings aim to contribute valuable insights into optimizing battery performance and addressing challenges associated with phase transformations, ultimately advancing the development of high-capacity anodes for lithium-ion batteries. REFERENCES [1] McDowell M. T., Lee S. W., Nix W. D., Cui Y. 25th Anniversary Article: Understanding the Lithiation of Silicon and Other Alloying Anodes for Lithium-Ion Batteries. Adv. Mater., Vol. 25 (36), pp. 4966–4985, 2013.[2] Chen L., Fan F., Hong L., Chen J., Ji Y. Z., Zhang S. L., Zhu T., Chen L. Q. A Phase-Field Model Coupled with Large Elasto-Plastic Deformation: Application to Lithiated Silicon Electrodes. J. Electrochem. Soc., Vol. 161 (11), pp. F3164–F3172, 2014.[3] Swaminathan, N., Balakrishnan, S., & George, K. Elasticity and Size Effects on the Electrochemical Response of a Graphite, Li-Ion Battery Electrode Particle. Journal of The Electrochemical Society, 163(3), A488–A498, 2016. Figure 1

Phase-Field Model of SEI Formation in Li-Metal Batteries

The formation of solid electrolyte interphase (SEI) is critical to the overall performance of lithium batteries. Here we have developed a rather comprehensive phase-field model for predicting the formation and growth of SEI, taking into account of multiple electrochemical/chemical reaction pathways leading to multiple SEI reaction products, diffusion of ionic and molecular species, and electron tunneling. All the SEI products are treated stoichiometric compounds without the typical approximation of parabolic composition dependence in almost all existing phase-field models involving stoichiometric compounds, allowing the direct connection between the phase-field model and atomistic calculations without further approximation. As a model system, we consider a 1-D prototypical half-cell with Li metal anode, liquid ethylene carbonate (EC) electrolyte with 1 molar (M) LiPF6, and SEI products Li2BDC and Li2CO3 under the open-circuit condition. The reduction potentials and activation barriers of the reactions are determined using atomistic simulations while the diffusivities of Li+ and EC in the bulk electrolyte are obtained by performing molecular dynamics simulations. The phase-field simulation is capable of predicting the evolution behavior of SEI products within a wide range of time scales from nanoseconds to a few seconds. It reveals that: (i) a porous Li2BDC layer forms at the anode/electrolyte interface and grows to a stable thickness of several tens of nanometers within ~20 μs; (ii) a porous Li2CO3 layer forms between anode and Li2BDC and grows to ~10 nm by partially consuming the existing Li2BDC layer within ~33 μs; and (iii) Li2CO3 forms a dense layer by filling the pores within ~20 ms. The stage (i) and (ii) are dominated by Li+ diffusion kinetics whereas stage (iii) is controlled by the reaction kinetics. The general computational framework can be extended and applied to the simultaneous evolution of Li-metal deposition and stripping and the SEI evolution.

Phase-Field Modeling of Zinc Dendrite Growth in Aqueous Zinc Batteries

The growth of zinc dendrites poses a major challenge in developing stable aqueous zinc batteries. It is crucial to understand the mechanism underlying the dendrite growth and the influences of various factors on the process of dendrite growth. In this study, a phase field model is constructed to simulate the growth of zinc dendrite and investigate the temporal and spatial distributions of ions and electric potential in both static and flowing electrolytes. The influences of various critical factors, including overpotential, anisotropic strength and electrolyte flow rate, on the morphology evolution of electrodeposit are investigated. Results reveal that the surface morphology evolution is intimately dependent on the competition between ion transport and the electrochemical reaction, and it can be modulated by various parameters. In particular, flowing electrolyte not only enhances the convective ion transport but also suppresses the dendrite bifurcation on the surface, resulting in a uniform distribution of the concentration gradient and a compact deposit. However, nonuniformity induced by the flowing electrolyte from the inlet to outlet may impede the stable cycling of zinc anodes in the flow batteries. Therefore, further design and optimization of the flow field are desired for zinc-based flow batteries. This work provides insights into the growth process of zinc dendrites and the influences of several critical parameters, which could inform the design of stable aqueous zinc batteries. Figure 1

Mechanical Failure of Core-Shell Cathode Particles: The Effects of Concentration-Dependent Material Properties and Phase Field Fracture Modelling

The use of core-shell cathode particles in lithium-ion batteries is an attractive approach to enhancing energy density whilst retaining lifetime, through reducing undesired reactions at the electrode-electrolyte interface and limiting the volume change of the electrode particles. However, mechanical failure through the fracture and debonding of the core-shell interface is a major challenge. In this work, we employ a coupled finite-element model to predict and mitigate the mechanical failure of core-shell cathode structures, taking as example a particle of NMC811 (core) coated with NMC111 (shell). In particular, we focus on two aspects:The first one involves the assumptions of material properties as inputs of the model. The material properties are often considered constant by battery modelling researchers, yet these parameters can vary significantly during charge/discharge. For example, experiments have unveiled a three-orders-of-magnitude drop in the diffusion coefficient of NMC materials during discharge [1]. Here, we incorporate material properties obtained from experimental data, including concentration-dependent diffusion coefficient obtained from GITT measurement and partial molar volume derived from in situ X-ray diffraction data. Our results indicate that when assuming a concentration-dependent partial molar volume, the maximum values of tensile hoop stress in the shell are nearly three times lower than those predicted with constant average properties, diminishing the likelihood of fracture.When accounting for concentration-dependent diffusion coefficient, large concentration gradient is observed near the outer surface of the core due to reduced lithium mobility at high states of lithiation, hindering full electrode capacity utilisation. The significant concentration gradient and capacity underutilisation align with direct observations from experiments [2]. Notably, these phenomena cannot be captured in modelling if assuming constant diffusion coefficient. These findings offer new perspectives on the performance and design of core-shell electrode particles. Safe design maps in terms of core size and shell thickness to mitigate fracture and debonding are obtained.The second focus of this work is the utilisation of phase field fracture model in predicting cracking behaviours within the core-shell particle. Phase field modelling provides a continuous representation of cracks, enabling the tracking of complex cracking patterns [3]. We predict cracking behaviors caused by volume changes during charge/discharge, as well as inaccessible capacity and newly created active surface area caused by cracking. Our multiphyics model can hopefully act as a catalyst, accelerating the development process of the core-shell structured cathode particles and shortening the time to market of advanced batteries with extended lifetime.