Phase-field Model Research Articles

Phase-field ductile fracture simulations of thermal cracking in additive manufacturing

We present a multiphysics phase-field fracture model for thermo-elasto-plastic solids in the context of finite deformation and apply it to simulate the hot cracking phenomenon during metal additive manufacturing. The model is derived in a thermodynamically consistent manner, with the intercoupling mechanisms among elastoplasticity, phase-field crack and heat transfer comprehensively considered. It involves particularly coupled parameters among these materials physics, e.g. plasticity-dependent degradation function and fracture toughness, damage-dependent yield surface and thermal properties, and temperature-dependent elastoplastic properties and fracture strength. The finite element implementation of the coupled phase-field model is benchmarked with simulation results of a tensile test of an I-shape specimen, encompassing elastoplasticity, hardening, necking, crack initiation and propagation, in contrast to the related experimental results. The validated model is further employed to simulate the multiphysics hot cracking phenomenon in additive manufacturing in the context of both the effective powder-bed model and the powder-resolved model thanks to prior non-isothermal phase-field powder-bed-fusion simulations. Simulation results reveal certain key features of the hot crack and its dependency on process parameters like beam power and scan speed, which are helpful for the fundamental understanding of crack formation mechanisms and process optimization.

Lattice Boltzmann simulation of plunging breakers

ABSTRACT In the present work, the conservative phase-field lattice Boltzmann model is applied to simulate plunging breakers in shallow waters. The solver is computationally efficient as it is explicit in time stepping and does not require a solution for the pressure Poisson equation. Nevertheless, the solver can simultaneously handle a large Reynolds number ( = 10 4 ) and a high-density ratio of 1000. Simulations are performed by varying initial steepness and dispersion parameters, which influence the nonlinearity degree of wave (or the growth speed of instability) and the oscillatory motion (limited by the bottom) on the breaking characteristics. Detailed analyses of kinematics, air entrainment, vorticity, and energy dissipation are presented.

Phase field modeling and numerical algorithm for two-phase dielectric fluid flows

We develop a method for modeling and simulating a class of two-phase flows consisting of two immiscible incompressible dielectric fluids and their interactions with imposed external electric fields in two and three dimensions. We first present a thermodynamically-consistent and reduction-consistent phase field model for two-phase dielectric fluids. The model honors the conservation laws and thermodynamic principles, and has the property that, if only one fluid component is present in the system, the two-phase formulation will exactly reduce to that of the corresponding single-phase system. In particular, this model accommodates an equilibrium solution that is compatible with the zero-velocity requirement based on physics. This property leads to a simpler method for simulating the equilibrium state of two-phase dielectric systems. We further present an efficient numerical algorithm, together with a spectral-element (for two dimensions) or a hybrid Fourier-spectral/spectral-element (for three dimensions) discretization in space, for simulating this class of problems. This algorithm computes different dynamic variables successively in an un-coupled fashion, and involves only coefficient matrices that are time-independent in the resultant linear algebraic systems upon discretization, even when the physical properties (e.g. permittivity, density, viscosity) of the two dielectric fluids are different. This property is crucial and enables us to employ fast Fourier transforms for three-dimensional problems. Ample numerical simulations of two-phase dielectric flows under imposed voltage are presented to demonstrate the performance of the method herein and to compare the simulation results with theoretical models and experimental data.

Numerical Voltammetry of Phase Separating Materials Using Phase Field Modeling

Higher capacity materials, such as Si and Sn are known to have phase separating behavior during the (de)lithiation. While initial models for lithiation in graphite electrode were based on single phase diffusion, with the introduction of Si and Sn, disposition of the models has shifted to the two-phase diffusion. It is important to understand the interaction of various phenomenon in materials which show phase change during (dis)charging cycles. In this work, we present a phase field model to simulate two-phase lithiation. This model is used to study the electrochemical response of the system by conducting numerical voltammetry. The main goal of this effort is to highlight the difference in electrochemical response occurring during single-phase diffusion and two-phase diffusion and explain the ensuing physics. Furthermore, effect of elasticity which governs the phase-change process and also alters corresponding voltammograms is also studied in detail. The voltammograms show clear shift in current peaks’ size and position for the changing diffusion behavior. Also as elasticity affects the two-phase diffusion, change in nucleation timing and diffusion rate are visible in voltammograms. Additionally, it is also observed how elasticity can cease the phase separation behavior and voltammogram for two-phase diffusion can become identical to single-phase diffusion.

Global Solutions to an Initial-Boundary Value Problem of a Phase-Field Model for Motion of Grain Boundaries

Global Solutions to an Initial-Boundary Value Problem of a Phase-Field Model for Motion of Grain Boundaries

Linear relaxation method with regularized energy reformulation for phase field models

In this paper, we establish a novel linear relaxation method with regularized energy reformulation for phase field models, which we name the RRER method. We employ the molecular beam epitaxy (MBE) model and the phase-field crystal (PFC) model as test beds, along with several coupled phase field models, to illustrate the concept. Our proposed RRER strategy is applicable to a broad class of phase field models that can be derived through energy variation. The RRER method consists of two major steps. First, we introduce regularized auxiliary variables to reformulate the original phase field models into equivalent forms where the free energy is transformed under the auxiliary variables. Then, we discretize the reformulated PDE model under these variables on a staggered time mesh for the phase field models. We incorporate the energy reformulation idea in the first step and the linear relaxation concept in the second step to derive a general numerical algorithm for phase field models that is linear and second-order accurate. Our approach differs from the classical invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, as we don't need to take time derivatives for the auxiliary variables. The resulting schemes are linear, i.e., only linear algebraic systems need to be solved at each time step. Rigorous theoretical analysis demonstrates that these resulting schemes satisfy the modified discrete energy dissipation laws and preserve the discrete mass conservation for the PFC and MBE models. Furthermore, we present numerical results to demonstrate the effectiveness of our method in solving phase field models.

Phase field simulation of dendrites morphology evolution in sodium metal batteries

Sodium (Na) dendrite growth poses challenges such as short circuits and capacity loss, and the underlying mechanisms are unclear. Although the experiments of Na dendrite growth were reported, there are very few simulations that can assist in analyzing its morphology evolution mechanism. In addressing this challenge, we develop a phase-field model to explore the influence of current density, anisotropic strength, and Na-ion consumption on the evolution of Na dendrite morphology. It is found that higher current density promotes more dendrite growth and lateral branches. Anisotropy strength influences dendrite growth rates horizontally and vertically, with horizontal growth exceeding or equaling vertical growth, reducing short-circuit risks. Increased Na-ion consumption results in significant lateral branches, kinks, and pores in dendrite structures. These results offer valuable insights into mitigating the formation of erratic dendrites, holding significant implications for the advancement of Na-based batteries with high stability and safety.

An adaptive dynamic phase-field modeling with variable-node elements for thermoelastic fracture in orthotropic media

In this work, a novel adaptive algorithm integrated into a dynamic phase-field method is developed to model the fracture of orthotropic materials under thermoelastic loading. The procedure of local refinement is synchronized with crack tip advancement, with the phase-field value serving as the criterion for marking elements. To address mesh refinement inconsistencies, variable-node elements are employed. Hughes–Hilbert–Taylor (HHT) and backward difference method are utilized for time discretization of displacement and temperature, respectively. A penalized structural matrix is incorporated into crack surface density to realize the orthotropy of crack growth. The performance of the proposed method is validated through several numerical benchmarks, i.e., the proposed method can effectively simulate the dynamic crack propagation of orthotropic materials under thermoelastic loading, and can assuage computational overhead while keeping acceptable accuracy.

A framework to model freeze/thaw-induced crack propagation in concrete based on a fatigue phase-field method

Concrete structures in cold regions are exposed to harsh environmental conditions, including freeze–thaw cycles, which lead to frost damage and degradation. Accurate simulation of the freeze–thaw damage process is essential for assessing concrete structures’ performance and service life. This paper presents a novel approach for simulating microcracks propagation in concrete during freeze–thaw cycles based on a fatigue phase-field model, in which a fatigue degradation function is introduced in the energy functional. The commercial software COMSOL is employed to implement the proposed model using a segregated scheme. The proposed model is validated against experimental results presented in the literature. Additionally, three cases are studied to get insight into the freeze–thaw damage. The results demonstrate the effectiveness of the fatigue phase-field model in capturing crack propagation in concrete under freeze–thaw cycles. The developed model offers valuable insights for assessing and designing concrete structures in cold regions.

Microstructure evolution during solidification in a low superheat casting process of the Al-Mg2Si composites having excess Si: A phase field study

A two-dimensional phase field (PF) model of solidification during low superheat casting has been developed to grab insight into the microstructure formation mechanism of the Al-xMg2Si-ySi composite, having extra Si as the grain refining agent. The developed PF model employs a seed undercooling based nucleation model to simulate the formation of primary Mg2Si and α-Al grains, wherein the interfacial energy of the Al-melt interface is taken from literature, and a Molecular Dynamics (MD) model is employed to calculate the interfacial energy of the Mg2Si-melt interface. The simulation study predicts the microstructural parameters such as grain size, and sphericity of the primary Al and primary Mg2Si phases, as well identifies the appropriate numerical parameters such as mobility, anisotropy parameters to study the microstructure evolution in the Al-xMg2Si-ySi composites. The kinetics of grain growth of both α-Al and primary Mg2Si have been studied. Novelty of the study lies in developing a near-accurate PF model capable of optimising the weight fraction of Mg2Si and excess Si in the low superheat cast Al-xMg2Si-ySi composite system, in view of obtaining the lower grain size and higher sphericity of both primary Al and primary Mg2Si grains, which is in line with the experimental observations. The proposed PF model is capable of predicting the faceted growth of Mg2Si particles in the Al-xMg2Si-ySi composite, in presence of extra Si and during low superheat casting, and the dendritic growth of Mg2Si in the binary Al-Mg2Si composite having high Mg2Si weight fraction. Following the Jackson’s model of interface growth, it has been demonstrated that the modification of kinetic coefficient leads to a transformation of morphology of Mg2Si particles from large dendritic to faceted ones.

A phase-field fracture model in thermo-poro-elastic media with micromechanical strain energy degradation

This work extends the hydro-mechanical phase-field fracture model to non-isothermal conditions with micromechanics based poroelasticity, which degrades Biot’s coefficient not only with the phase-field variable (damage) but also with the energy decomposition scheme. Furthermore, we propose a new approach to update porosity solely determined by the strain change rather than damage evolution as in the existing models. As such, these poroelastic behaviors of Biot’s coefficient and the porosity dictate Biot’s modulus and the thermal expansion coefficient. For numerical implementation, we employ an isotropic diffusion method to stabilize the advection-dominated heat flux and adapt the fixed stress split method to account for the thermal stress. We verify our model against a series of analytical solutions such as Terzaghi’s consolidation, thermal consolidation, and the plane strain hydraulic fracture propagation, known as the KGD fracture. Finally, numerical experiments demonstrate the effectiveness of the stabilization method and intricate thermo-hydro-mechanical interactions during hydraulic fracturing with and without a pre-existing weak interface.

Improving lithium deposition in porous electrodes: Phase field simulation

The development of structured lithium metal anodes is a key area of focus in the field of lithium battery research, which can significantly improve the energy density, cycle life and safety of lithium metal batteries. In this study, an electrochemical phase field model is used to construct the total free energy of the electrochemical system in conjunction with the effect of the porous electrode scaffold. It investigates the anisotropy of the diffusion coefficient of lithium ions in the electrolyte and the effect of the gradient size of the porous electrode scaffolds on the lithium deposition within the pores. Our numerical results show that increasing the diffusion coefficient of lithium ions in the x-direction cannot improve the lithium deposition within the pores. Instead, it leads to the appearance of lithium dendrites in the top layer of the porous electrode. Appropriately increasing the diffusion coefficient of lithium ions in the y-direction can significantly increase the lithium deposition capacity within the pores. However, excessive increases can lead to a reversal of the lithium deposition capacity within the pores. In addition, adopting the gradient size of the porous electrode scaffold in the y-direction can effectively alleviate the phenomenon that lithium metal is preferentially deposited at the top of the scaffold. These findings provide new ideas for the future design of structured lithium metal anode batteries.

Conducting viscous bodies with phase transitions: Deriving Gurtin’s postulate from the second law structure invariance

We prove that the basic structures of phase-field models for phase-transitions in non-isothermal setting can be derived all together from a unique principle requiring structure invariance of the second law of thermodynamics, written as a Clausius-Duhem inequality, under orientation-preserving diffeomorphism-based changes of observer and standard regularity conditions. We thus prove that the independent scalar balance of actions driving phase transitions does not require to be postulated, as it was in Gurtin’s 1996 proposal, rather it is a natural consequence of the invariance requirement adopted and maintains independence from constitutive choices. The approach discussed here is also a model-building framework for the dynamics of complex bodies and allows one to show that the microstructural behavior can be responsible of effects leading to wave-type heat propagation. The results indicate another role for the second law of thermodynamics, in addition to being a source of constitutive restrictions and compatibility or stability conditions.

Numerical modeling of shear band effect on Goss grain recrystallization in electrical steels: Crystal plasticity finite element and phase field modeling

This study investigates the effect of shear band evolution on the nucleation of Goss {110}<001> texture during the primary recrystallization of 3.24 wt% Si grain-oriented electrical steel. Nucleation at the early stage of primary recrystallization of the steel is explored both experimentally and numerically. The experimental approach involves cold rolling the steel specimens to obtain a thickness reduction ratio of 76 % and then applying heat treatment to them at 600 °C for less than 1 min. The numerical simulation employes crystal plasticity (CP) finite element model (FEM) to simulate the plastic deformation induced by the dislocation slips on predefined slip systems and non-crystallographic shear bands during cold rolling. Based on the CPFEM results, the generalized strain energy release maximization (GSERM) model is used to predict the preferential orientation probability of recrystallized nuclei for the steel by considering shear band formation. Subsequently, the microstructure evolution during the early stage of primary recrystallization of the steel is simulated using the phase field model (PFM). The developed CP model successfully predicted shear band activation and evolution in the γ-fibers centered on the {111}<112> texture component. The model also demonstrated that shear bands would be the preferred nucleation sites at the early stage of primary recrystallization because of their high stored energy. Moreover, by coupling with the GSERM model, the PFM could reproduce the nucleation of Goss grains at the beginning of primary recrystallization in shear bands.

Phase-field simulation study on dendritic growth behavior during bilateral directional solidification

The directional solidification technique can uniform the crystal growth direction within the material, but does not remove the lateral side branches of the axial main arm during the growth process. The influence of such lateral side branches existing in the directionally solidified dendritic organisation on the mechanical properties of the material cannot be neglected. In this paper, based on the Kim-Kim-Suzuki (KKS) phase-field model, the competitive behaviour of lateral secondary dendrites between the two main arms under different boundary conditions and non-synchronous growth conditions is investigated for Fe-0.5 %C alloys. In addition, the diffusion and enrichment rules of solutes are analysed by the distribution of solutes. The results show that the molten pool is spherical without external force, and the size distribution of the molten pool is as follows: dendrite tip collision zone > dendrite root > dendrite wall, and the solute segregation rule is the same; The spacing of the main arms has little effect on the concentration values between the dendrites, and the solutes are very symmetrically over the entire computational region; The dendrite tip interface concentration during the stable phase of dendrite growth is 0.89 %. Without considering the initial liquid-phase concentration, the concentration in the collision zone at the dendrite tip is approximately twice the concentration at the tip of the dendrite in the stable period, which is 1.28 %, when the bilateral dendrites are encountered. Dendrites always collide preferentially on the centreline of the included angle under non-parallel conditions, and solute-enriched zones also occur on the centreline of the included angle; During the curved surface bilaterally directional solidification process, when the left and right sides have the same shape of surfaces, the solute enrichment at the end of solidification is on the centreline. For the case of a single planar boundary, the solute-enriched region is influenced by the curved surface and is similar to the initial curved surface profile.

A modified phase-field model simulating multiple cracks propagation of fissured rocks under compressive or compressive-shear conditions

The Phase-Field Method (PFM) is a highly effective tool for predicting crack propagation in solid materials. In this study, we propose a modified phase-field model tailored for simulating the propagation of multiple cracks in fissured rocks. The model incorporates a novel approach by decomposing the crack driving energy into tension, tensile-shear, and compressive-shear components. Moreover, it employs a staggered algorithm scheme to solve the partial differential equations using the finite element method (FEM). The accuracy of the model is further validated through benchmark tests of uniaxial compression and pure shear, demonstrating exceptional ability to predict compressive-shear behavior in intermittent fissured rocks. Additionally, we successfully simulate various hole shapes and investigate the pressure-sensitive effect on multiple crack propagation. Overall, the proposed model extends the applicability of the phase-field approach and shows promise for use in underground rock engineering applications.

Phase-field model of glass transition: behavior under uniform quenching

ABSTRACT The process of transition from the liquid to the solid state during the uniform quenching is described by a combination of two theoretical approaches: the phase field method and the gauge theory, which suggests the phase transition in the subsystem of topologically stable distortions. Using the stochastic partial differential equations we numerically simulate competition between the processes of crystallization and vitrification during the rapid cooling of the melt. A glass order parameter which is suitable for the characterization of transition to the glassy state has been proposed. At low cooling rates, the domain freezes into the crystalline structure with defects, and its glass order parameter is orders of magnitude less than that for the vitrified state obtained by rapid quenching. Proposed method improves and extends the theoretical description of glass transition, qualitatively represents the basic kinetic properties of the process and demonstrates sensitivity of resulting microstructures to the quenching rate.

Drop Behavior on Heterogeneous Ratchet-Structured Substrates Harmonically Vibrated in Lateral Direction.

We analyze numerically a new ratchet system: a liquid drop is sitting on a heterogeneous ratchet-structured solid plate. The coated plate is subject to a lateral harmonic oscillation. The systematic investigation performed in the frame of a phase field model shows the possibility of realizing a long-distance net-driven motion for isolated domains of the forcing parameters. The studied problem might be of considerable interest for controlled motion in micro- and nanofluidics.

Characterizing pearlite transformation in an API X60 pipeline steel through phase-field modeling and experimental validation

This study explores the microstructural characterization of pearlite phase transformation in high-strength low-alloy API X60 steel, which is used in pipelines. Understanding the formation, phase percentages, and morphology of the pearlitic phase is crucial since it affects the mechanical properties of the considered steel. In this research, a phase-field model, particularly the Cahn–Hilliard approach, was used in order to simulate the formation and morphology of the pearlite phase in response to different heat treatments. Both double- and triple-well potentials were considered for comprehensively studying pearlite’s morphology in the simulations. The simulation results were then compared with experimental outcomes obtained by metallography and field-emission scanning electron microscopy analyses. Considering the double-well potential can help simulate only two phases, ferrite and cementite, which is less compatible with the experiment results than the triple-well potential, which gives the possibility of simulating a three-phase microstructure, ferrite, cementite, and austenite, and a better match with experimental data. The study revealed that as the cooling rate increases, the interlamellar spacing and layer thickness decrease. Additionally, the difference between experimental and simulation results using triple-well potential was approximately ∼10%. Therefore, triple-well potential formulation predictions have better agreements with experimental results for the development circumstance of pearlitic structures.

Phase-field modelling of dynamic hydraulic fracturing in porous media using a strain-based crack width formulation

This paper presents a comprehensive study on phase-field modelling in COMSOL MultiPhysics for simulating dynamic hydraulic fracturing in porous media based on Biot’s poro-elasticity theory. The focus is on addressing the challenges associated with crack width estimation in this context. A new strain-based crack width formulation is proposed, offering improved accuracy in predicting fracture permeability and ease of implementation in numerical approaches. The model’s capabilities are extended to consider dynamic crack propagation by incorporating the kinetic energy in the governing coupled hydro-mechanical-damage equations. The numerical implementations in COMSOL MultiPhysics are thoroughly explained, providing insights into the techniques used to solve the governing equations. Verification examples, including the benchmark KGD verification, are presented to demonstrate the model’s capabilities in simulating hydraulic fractures in porous media and validate its accuracy and reliability. A final numerical example focusing on the dynamics of crack propagation in a gravity dam is simulated, allowing for a comprehensive examination of the model’s performance. The proposed strain-based crack width formulation and consideration of dynamic crack propagation contribute to improved accuracy in predicting fracture permeability.