Phase-field Model Research Articles

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.

Order evolution from a high‐entropy matrix: Understanding and predicting paths to low‐temperature equilibrium

AbstractInterest in high‐entropy inorganic compounds originates from their ability to stabilize cations and anions in local environments that rarely occur at standard temperature and pressure. This leads to new crystalline phases in many‐cation formulations with structures and properties that depart from conventional trends. The highest‐entropy homogeneous and random solid solution is a parent structure from which a continuum of lower‐entropy offspring can originate by adopting chemical and/or structural order. This report demonstrates how synthesis conditions, thermal history, and elastic and chemical boundary conditions conspire to regulate this process in Mg0.2Co0.2Ni0.2Cu0.2Zn0.2O, during which coherent CuO nanotweeds and spinel nanocuboids evolve. We do so by combining structured synthesis routes, atomic‐resolution microscopy and spectroscopy, density functional theory, and a phase field modeling framework that accurately predicts the emergent structure and local chemistry. This establishes a framework to appreciate, understand, and predict the macrostate spectrum available to a high‐entropy system that is critical to rationalizing property engineering opportunities.

A convergence result for the derivation of front propagation in nonlocal phase field models

We prove that the mean curvature of a smooth surface in \R^{n} , n\geq 2 , arises as the limit of a sequence of functions that are intrinsically related to the difference between an n - and 1 -dimensional fractional Laplacian of a phase transition. Depending on the order of the fractional Laplace operator, we recover the fractional mean curvature or the classical mean curvature of the surface. Moreover, we show that this is an essential ingredient for deriving the evolution of fronts in fractional reaction-diffusion equations such as those for atomic dislocations in crystals.

Simulation of Dendrite Remelting via the Phase-Field Method

The solidification of alloys is a key physical phenomenon in advanced material-processing techniques including, but not limited to, casting and welding. Mastering and controlling the solidification process and the way in which microstructure evolution occurs constitute the key to obtaining excellent material properties. The microstructure of a solidified liquid metal is dominated by dendrites. The growth process of these dendrites is extremely sensitive to temperature changes, and even a small change in temperature can significantly affect the growth rate of the dendrite tip. Dendrite remelting is inevitable when the temperature exceeds the critical threshold. In this study, a temperature-induced-dendrite remelting model was established, which was implemented through the coupling of the phase field method (PFM) and finite difference method (FDM). The transient evolution law of dendrite remelting was revealed by simulating dendritic growth and remelting processes. The phase field model showed that the lateral dendrites melt first, the main dendrites melt later, and the main dendrites only shrink but do not melt when the lateral dendrites have not completely melted or the root is not broken. The long lateral branches break into fragments, while the short lateral branches shrink back into the main dendrites. The main dendrites fracture and melt in multiple stages due to inhomogeneity.

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.

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.

Phase Field Modeling of Hydraulic Fracturing with Length-Scale Insensitive Degradation Functions

A length-scale insensitive degradation function is applied to extend the cracks during hydraulic fracturing under stress boundary conditions in this study. The phase field method is an effective modeling technique that has great potential for use in hydraulic fracturing. Nonetheless, current hydraulic fracturing research is still concentrated on small scales. The phase field model employs a degradation function that is insensitive to length scale, allowing for the decoupling of the phase field length scale from the physical length scale. This facilitates the simulation of hydraulic fracturing crack extensions in larger structures with a consistent mesh density. The correctness of the phase field method is verified firstly by comparing with the experimental results, and the accuracy and efficiency of the proposed method are further verified through a series of numerical calculations.

Phase Field Coupled Finite Deformation Plasticity Formulation of Ductile Fracture With Nonlinear Kinematic Hardening and Modified Energy Release Function

ABSTRACTA ductile damage theory is presented by coupling the covariant formulation of finite deformation plasticity with the phase field modeling of fracture, including kinematic hardening for the ductile response of the materials. A phase field coupled nonlinear kinematic hardening equation is proposed in the reference configuration having equivalent representation through the Lie derivative of the kinematic hardening tensor by push‐forward operation in the spatial configuration, thus ensuring the satisfaction of frame invariance. To capture the correct physical response of the material by the phase field evolution equation, the fracture driving function, that is, the difference between the sum of elastic and plastic energies and threshold energy, after the damage initiation is modified by an energy release controlling function, which is an empirical relation of equivalent plastic strain. In defining the energy release controlling function, well‐defined points with physical significance in the experimental load versus displacement curve are used. To simulate the response of the material in relatively large time steps, a modified staggered scheme is presented, evaluating the fracture driving and energy release controlling functions from the previous converged step and using the updated phase field variable in the weak form of the momentum balance equation. To quantify different material parameters from available experimental results in the literature, the developed phase field coupled elasto‐plastic model uses a neural network optimization procedure consisting of neural network training together with optimization in MATLAB and finite element model evaluation in Abaqus user element subroutine UEL. Model capabilities are demonstrated by simulating the crack propagation in complex 3D geometries such as the second and third Sandia Fracture Challenges.

Creep rafting fracture and strain properties of overheating Co-based superalloy: Crystal plasticity phase-field simulation

The creep fracture and rafting of Co-based superalloys under thermal loading are critical properties for the long time working at high temperature, the crystal plasticity phase-field (CPPF) model is developed by incorporating the creep damage theory. The study of creep in Co-5Al-14V (at.%) superalloy in states of isothermal, continuous heating and overheating is realized. The rafting mechanism and evolution kinetics of γ′–Co3(Al, V) phase are analyzed in processes of directional coarsening and dissolution fracture under thermal loading stress, the incomplete P-type rafting morphology with low volume fraction of γ′ phase shows a reduced creep resistance. The creep strain has a steep rise for the rafting fracture and degeneration induced by overheating stress. The overheating followed by cooling triggers the recovery of γ′ rafts and improves the creep resistance. In addition, the dissolution and fracture of γ′ rafts are driven by the chemical potential, while the rafting and re-precipitation of γ′ phase are driven by the elastic potential under the creep stress. At the tertiary creep stage, the damage variable and creep strain increase in the order of isothermal creep, overheating and continuous heating. The studies show the creep fracture and strain properties in stress-diffusion mechanisms by coupling experiment morphology and CPPF simulation, the relationship of creep fracture and rafting properties of superalloys is established.

Evolutionary Mechanism of Solidification Behavior in the Melt Pool during Disk Laser Cladding with 316L Alloy

Laser cladding is an emerging environmentally friendly surface-strengthening technology. During the cladding process, the changes in molten pool temperature and velocity directly affect the solidification process and element distribution. The quantitative revelation of the directional solidification mechanism in the molten pool during the cladding process is crucial for enhancing the quality of the cladding layer. In this study, a multi-field coupling numerical model was developed to simulate the coating process of 316L powder on 45 steel matrices using a disk laser. The instantaneous evolution law of the temperature and flow fields was derived, providing input conditions for simulating microstructure evolution in the molten pool’s paste zone. The behavior characteristics of the molten pool were predicted through numerical simulation, and the microstructure evolution was simulated using the phase field method. The phase field model reveals that dendrite formation in the molten pool follows a sequence of plane crystal growth, cell crystal growth, and columnar crystal growth. The dendrites can undergo splitting to form algal structures under conditions of higher cooling rates and lower temperature gradients. The scanning speed of laser cladding (6 mm/s) has minimal impact on dendrite growth; instead, convection within the molten pool primarily influences dendrite growth and tilt and solute distribution.

Impact of Secondary‐Phase Particle Morphology on Grain Growth in Pure Copper Using a Phase Field Simulation Approach

Microstructural stability of polycrystalline alloys depends on the motion of grain boundaries during heat treatment. The presence of second‐phase particles (SPP) is one of the most effective mechanisms to stabilize the polycrystalline structure by pinning grain boundaries. The current study utilizes phase field modeling to determine the impact of particle size, morphology, and number distribution on grain growth in a face centered cubic copper system. The simulation results indicate that a distinct relationship exists between grain boundary motion and particles of different shapes. Above a minimum area fraction threshold for SPP, the number of particles per unit area emerges as the primary factor controlling the rate of grain growth. Particle shape aspect ratio shows influence on pinning grain boundaries as well with elongated particle shapes demonstrating improved grain boundary pinning, especially when oriented with the long axis parallel to the migrating grain boundary. The results presented here are in good agreement with experimentally observed trends, and the validation of a simplified phase field model allows for added complexity in future simulations.

An electro-thermo-mechanical coupling phase-field model of defect evolution induced by electromigration in interconnects

In this paper, the defect evolution caused by electromigration induced surface diffusion in interconnects is investigated using a newly-developed electro-thermo-mechanical coupling phase-field model. The Joule heat and its resulting thermomigration are included into the phase-field model. The governing equation of the phase-field is solved by semi-implicit spectral methods and the accompanied governing equations of applied physics fields are solved by finite volume methods. Comparative investigation into defect evolution with and without the influence of Joule heating is conducted. It is deduced that thermomigration facilitates local elongation of the defect in the “current crowding” region and exerts a substantial influence on the defect morphological evolution. Subsequently, the effect of the inclination angle of the electric field on the void morphology evolution and crack propagation is discussed. We find that the defect achieves the largest characteristic length when the electric field direction is perpendicular to the uniaxial tension direction, implying a higher threat to the circuit safety. This study may help to deepen people's understanding of how the thermal effect functions in electromigration process and sheds light on different modes of defect evolution in interconnects.

Dynamic thermal shock resilience of functionally graded materials: An adaptive phase-field approach

This work presents a phase field model for dynamic fracture under thermo-mechanical loads to analyze functionally graded material (FGM). Coupled governing equations are derived by minimizing the total energy consisting of kinetic energy and elastic strain energy, along with the energy due to the external heat source. To overcome the computational expense associated with the standard finite element analysis for implementing the phase field model, an adaptive approach based on the quadtree algorithm is employed that enhances computational efficiency while preserving accuracy. The hanging nodes within the element are treated as polygonal elements. The governing equations are solved using a staggered solution algorithm, following a hybrid phase field implementation strategy. The notch location, temperature variation, and gradation profile of FGM are carefully examined and investigated using numerous examples. It is observed that these parameters significantly influence the behavior of FGM. The results align well with the existing literature, validating our methodology within the examples explored. Additionally, the computational efficiency of adaptive meshing techniques is assessed, demonstrating the significant reductions in computational overhead while preserving accuracy and versatility in handling multi-physics fracture scenarios.

Effect of molybdenum addition on precipitate coarsening kinetics in Inconel 740H: A phase-field study

Inconel 740H (IN740H) has emerged as an important candidate for the advanced ultra-supercritical (AUSC) steam turbines due to its superior microstructural stability and creep resistance under service conditions. In the present work, we have reparameterized a multicomponent Ni-based superalloy (IN740H) into an equivalent ternary Ni-Al-Mo superalloy, based on the partitioning coefficients of the elements, using the thermodynamic database (CALPHAD). We use this thermodynamic description to employ a quantitative phase-field model to assess the long-term stability of γ′ precipitates in IN740H utilizing GPU-based supercomputing architecture. The assessment helps us to enhance our understanding of the effect of the atomic diffusivity of Mo on coarsening kinetics of the γ′-precipitates in equivalent ternary Ni-Al-Mo superalloy. Investigation reveals that our phase-field model can accurately predict the experimentally observed coarsening kinetics in IN740H.

An adaptive acceleration scheme for phase-field fatigue computations

Abstract Phase-field models of fatigue are capable of reproducing the main phenomenology of fatigue behavior. However, phase-field computations in the high-cycle fatigue regime are prohibitively expensive due to the need to resolve spatially the small length scale inherent to phase-field models and temporally the loading history for several millions of cycles. As a remedy, we propose a fully adaptive acceleration scheme based on the cycle jump technique, where the cycle-by-cycle resolution of an appropriately determined number of cycles is skipped while predicting the local system evolution during the jump. The novelty of our approach is a cycle-jump criterion to determine the appropriate cycle-jump size based on a target increment of a global variable which monitors the advancement of fatigue. We propose the definition and meaning of this variable for three general stages of the fatigue life. In comparison to existing acceleration techniques, our approach needs no parameters and bounds for the cycle-jump size, and it works independently of the material, specimen or loading conditions. Since one of the monitoring variables is the fatigue crack length, we introduce an accurate, flexible and efficient method for its computation, which overcomes the issues of conventional crack tip tracking algorithms and enables the consideration of several cracks evolving at the same time. The performance of the proposed acceleration scheme is demonstrated with representative numerical examples, which show a speedup reaching up to four orders of magnitude in the high-cycle fatigue regime with consistently high accuracy. Graphical abstract

Dendrite Growth and Dead Lithium Formation in Lithium Metal Batteries and Mitigation Using a Protective Layer: A Phase-Field Study.

Lithium metal batteries (LMBs) are considered one of the most promising next-generation rechargeable batteries due to their high specific capacity. However, severe dendrite growth and subsequent formation of dead lithium (Li) during the battery cycling process impede its practical application. Although extensive experimental studies have been conducted to investigate the cycling process, and several theoretical models were developed to simulate the Li dendrite growth, there are limited theoretical studies on the dead Li formation, as well as the entire cycling process. Herein, we developed a phase-field model to simulate both electroplating and stripping process in a bare Li anode and Li anode covered with a protective layer. A step function is introduced in the stripping model to capture the dynamics of dead Li. Our simulation clearly shows the growth of dendrites from a bare Li anode during charging. These dendrites detach from the bulk anode during discharging, forming dead Li. Dendrite growth becomes more severe in subsequent cycles due to enhanced surface roughness of the Li anode, resulting in an increasing amount of dead Li. In addition, it is revealed that dendrites with smaller base diameters detach faster at the base and produce more dead lithium. Meanwhile, the Li anode covered with a protective layer cycles smoothly without forming Li dendrite and dead Li. However, if the protective layer is fractured, Li metal preferentially grows into the crack due to enhanced Li-ion (Li+) flux and forms a dendrite structure after penetration through the protective layer, which accelerates the dead Li formation in the subsequent stripping process. Our work thus provides a fundamental understanding of the mechanism of dead Li formation during the charging/discharging process and sheds light on the importance of the protective layer in the prevention of dead Li in LMBs.

Parametric sensitivity analysis of the modified phase-field model in polymer membrane formation by immersion precipitation method

Parametric sensitivity analysis of the modified phase-field model in polymer membrane formation by immersion precipitation method

Phase-field approach for precise fracture tracking in anisotropic rocks: Integrating orthotropy-based energy decomposition and two-fold symmetric fracture toughness

Rock formations are known to exhibit material anisotropy, both in terms of elastic and fracture properties. This means that the fracture path in such formations is not a priori known but rather a complex unknown that requires robust numerical techniques to predict accurately. In this context, the phase-field model is considered particularly effective, provided that certain physical considerations are carefully adjusted to align with the physics of the problem. While addressing elastic anisotropy is well-established, the tension–compression asymmetry necessary to inhibit crack interpenetration in phase-field fracture models needs to account for the specific material anisotropy. Additionally, to accurately capture crack propagation, it is critical to simultaneously account for orientation-dependent fracture toughness in such materials. To address this, the present study employs an anisotropic phase-field model that integrates the generalized spectral decomposition proposed in the literature for orthotropic materials with a two-fold symmetric fracture toughness, to predict the fracture trajectories in rock-type samples under fixed mixed-mode loading ratios. While each of the two aspects has primarily been applied to model orthotropic plates under simple tensile and shearing loading conditions in the literature, here we study their applicability in complex loading scenarios. To this end, the experimental data from notched semi-circular specimens of Grimsel Granite undergoing complex mixed-mode loading obtained in our previous work is considered. We focus on two given mode-mixity ratios and perform numerical studies. Our results emphasize the importance of considering this generalized decomposition for phase-field modeling of fracturing in rock-type materials, particularly under loading conditions where the crack might otherwise be unrealistically driven into the compressive region. Although certain features are well captured by considering anisotropy in elasticity alone, our findings demonstrate that incorporating a two-fold symmetric fracture toughness proves to be advantageous for more precise tracking of the fracture path.

An improved phase-field model for fatigue crack growth considering constraint effects

The phase field model is a promising method for simulating fatigue crack growth (FCG) behavior. However, the conventional phase field (PF) model may not adequately account for constraint effects, where fracture toughness is affected by geometries. Therefore, stress triaxiality is incorporated into the PF model by modifying the fracture energy release rates to consider constraint effects. The model successfully simulates the FCG behavior of different geometries, such as CT, SENB, and MT specimens, as well as the mixed-mode FCG behavior of CTS specimens and other complex geometries. All simulations agree well with experiments, proving that our model is capable to capture the constraint effects in FCG behavior. These findings indicate that stress triaxiality is important to capture the constraint effects.

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.