Phase-field Model Research Articles

An improved phase-field model for oil–water two-phase flow and mixed-mode fracture propagation in hydraulic fracturing

An improved novel phase-field model is proposed for describing fracture propagation, effectively characterizing the interactions between oil–water two-phase fluids and solids as well as mixed-mode fracturing. This model encompasses equations for two-phase flow stress balance, fluid flow, and a complex mode fracture phase-field specific to two-phase flow. Within the fluid flow equations, the capillary pressure caused by the oil–water interface during the fluid loss process in the matrix is accounted for, and the anisotropic relative permeability during fluid flow is linked to normalized saturation. The driving forces for fracture propagation in the phase-field are categorized into Mode I and Mode II forces, with the contribution from the oil–water two-phase fluid attributed to the Mode I (tensile) driving force. The model uses finite element numerical discretization and the Newton-Raphson iterative method to establish a numerical iteration format, employing an implicit-explicit staggered solution scheme. Additionally, the model is used to investigate several key factors. It assesses the impact of different intersection angles on fracture propagation in naturally porous media. It also studies the effect of different natural fracture permeabilities on hydraulic fracture propagation. Finally, the model analyzes the propagation patterns of hydraulic fractures under different perforation phase angles.

Formation mechanism of radial and circular cracks promoted by delamination in drying silica colloidal deposits.

Cracks with radial and circular patterns are appealing in nature and industry. Although morphologies and propagation conditions of cracks are extensively studied, the formation mechanism of crack pattern by the interaction of channel fracture and interfacial delamination remains elusive. Here, we present the transition of radial to coexisting radial and circular crack patterns when the thickness of colloidal deposits on both hard and soft substrates exceeds a critical value, through the colloidal volume fraction dependence. In addition, a thickness-dependent phase diagram from radial crack to coexistence of radial and circular cracks was constructed with respect to the radius and the volume fractions of silica colloidal deposits. A phase-field fracture model is developed to elucidate how the formation of radial cracks is facilitated by simultaneous delamination. The warping-induced radial tensile stress at the bottom surface of the striped deposit is proportional to the thickness. It leads to subsequent nucleation and growth of circular cracks in thick deposits. This work provides insight into the formation mechanism of complex crack patterns in drying colloidal deposits and revolutionizes the design space of crack-based micro-nano structures.

Phase-field-based lattice Boltzmann method for containerless freezing.

In this paper we first propose a phase-field model for the containerless freezing problems, in which the volume expansion or shrinkage of the liquid caused by the density change during the phase change process is considered by adding a mass source term to the continuum equation. Then a phase-field-based lattice Boltzmann (LB) method is further developed to simulate solid-liquid phase change phenomena in multiphase systems. We test the developed LB method by the problem of conduction-induced freezing in a semi-infinite space, the three-phase Stefan problem, and the droplet solidification on a cold surface, and the numerical results are in agreement with the analytical and experimental solutions. In addition, the LB method is also used to study the rising bubbles with solidification. The results of the present method not only accurately capture the effect of bubbles on the solidification process, but also are in agreement with the previous work. Finally, a parametric study is carried out to examine the influences of some physical parameters on the sessile droplet solidification, and it is found that the time of droplet solidification increases with the increase of droplet volume and contact angle.

Efficient numerical scheme with full decoupling, second-order temporal accuracy and unconditional energy stability for a flow-coupled melt-convective dendritic solidification phase-field model

Efficient numerical scheme with full decoupling, second-order temporal accuracy and unconditional energy stability for a flow-coupled melt-convective dendritic solidification phase-field model

A numerical model for solitary wave breaking based on the phase-field lattice Boltzmann method

This study presents a numerical investigation of a solitary wave breaking over a slope by using the phase-field lattice Boltzmann method. The incompressible two-phase flow equations are solved by using a velocity-based formulation of the two-phase lattice Boltzmann method with a central-moment collision model to accurately simulate wave breaking problems. For interface capture, a phase-field lattice Boltzmann method that ensures mass conservation is employed. The validity of the proposed method is confirmed through solitary wave propagation and transformation problems, and the obtained results are in good agreement with the experimental and calculated results. The proposed method is then employed to analyze wave breaking on a slope, demonstrating strong concordance with experimental data and existing computational findings. By analyzing the instantaneous flow characteristics and the temporal evolution of the variation in kinetic, potential, and total energy from deep to shallow water, the model can reveal the macroscopic characteristics of solitary wave breaking. Because the phase-field model effectively simulates wave breaking and air entrainment, it can depict wave energy dissipation more accurately than the single-phase lattice Boltzmann method with free surface tracking.

Phase-field modeling for studying bond mechanisms in pull-out tests of corroded reinforced concrete structures

Phase-field modeling for studying bond mechanisms in pull-out tests of corroded reinforced concrete structures

Sparse polynomial chaos expansion and adaptive mesh refinement for enhanced fracture prediction using phase-field method

Phase-field models (PFMs) have proven to accurately predict complex crack patterns such as crack branching, merging, and crack fragmentation, but they are computationally costly. This study stems from the high computational costs associated with non-adaptive PFMs for brittle fracture analysis, which require dense meshes. This makes training data generation for surrogate models prohibitively expensive. To address these challenges, this paper proposes an adaptive mesh refinement (AMR) informed sparse polynomial chaos expansion (PCE) framework. The AMR algorithm is incorporated into the fracture analysis workflow to maximize the computational efficiency in data generation, eliminating the need for pre-refinement. The AMR informed sparse PCE is calibrated with Bayesian compressive sensing (BCS) to efficiently map domain input parameters to quantities of interest (QoIs), even with limited training data. We employ a multi-output QoI approach to predict load–displacement relationships with high resolution. The novelty of this work lies in the integration of AMR into the PFM workflow and the application of sparse PCE for brittle fracture analysis. Benchmark tests demonstrate the proposed method’s superior accuracy and computational efficiency compared to conventional AMR method, marking a notable improvement in fracture mechanics.

Machine learning surrogate for 3D phase-field modeling of ferroelectric tip-induced electrical switching

Phase-field modeling offers a powerful tool for investigating the electrical control of the domain structure in ferroelectrics. However, its broad application is constrained by demanding computational requirements, limiting its utility in inverse design scenarios. Here, we introduce a machine-learning surrogate to accelerate 3D phase-field modeling of tip-induced electrical switching. By dynamically handling the boundary conditions, the surrogate achieves accurate reproduction of switching trajectories under various tip locations and applied voltages. With stable predictions throughout entire morphological evolution pathways and a relative error inferior to 10% compared to direct solvers, the model efficiently emulates intricate switching sequences. By successfully replicating the boundary conditions, the presented framework strides towards a holistic surrogate for the ferroelectric phase field. With up to 2500-fold speed-ups over classical methods, our approach opens the path for the tractable design of the domain structure and the resolution of realistic inverse problems.

Microstructure forming mechanism of inconel 625 alloy fabricated by laser/ultra-high (UHF) induction hybrid deposition method

To reveal the microstructure forming mechanism of laser/ultra-high frequency (UHF) induction deposition, this paper developed a microscopic phase-field (PF) model to numerically investigate dendrite growth during solidification. The macroscopic model of molten pool evolution is adopted to provide the solidification conditions for the microscopic PF model. The dendrite growth during laser deposition is simulated to evaluate the effect of UHF induction heat on the dendrite growth. Results show that because of the high temperature gradient and cooling rate, the PDAS of laser-UHF induction hybrid deposited layer is less than that of the laser deposited layer. The UHF induction heat also leads to a high flow velocity of the molten metal during laser-UHF induction hybrid deposition. The high flow velocity contributes to the decrease in PDAS by inhibiting the interdendritic enrichment of solute. During laser-UHF induction hybrid deposition, a higher solute gradient is present in the tip region of dendrite arm, leading to a faster dendrite growth rate. The UHF induction heat also increases the solute distribution coefficient during deposition, which further inhibits the element segregation. Under the action of UHF induction heat, a low interdendritic solute gradient and an evenly distributed solute can be obtained, thus helping increase interdendritic undercooling degrees and decreasing the PDAS. The simulated PDAS and solute distribution have good consistency with the experimental results. The spectral analysis of EDS line detection indicates that the laser-UHF induction hybrid deposited layer has a more refined microstructure and weaker element segregation than the laser deposited layer does.

Estimation of Boil-Off Gas Generation by using Phase-Field Model for Liquid Hydrogen Vacuum-Jacketed Valve

Estimation of Boil-Off Gas Generation by using Phase-Field Model for Liquid Hydrogen Vacuum-Jacketed Valve

A generalized phase-field cohesive zone model ([formula omitted]PF-CZM) for fracture

In this work a generalized phase-field cohesive zone model (μPF-CZM) is proposed within the framework of the unified phase-field theory for brittle and cohesive fracture. With the introduction of an extra dissipation function for the crack driving force, in addition to the geometric function for the phase-field regularization and the degradation function for the constitutive relation, theoretical and application scopes of the original PF-CZM are broadened greatly. These characteristic functions are analytically determined from the conditions for the length scale insensitivity and a non-shrinking crack band in a universal, optimal and rationalized manner, for almost any specific traction–separation law. In particular, with an optimal geometric function, the crack irreversibility can be considered without affecting the target traction–separation softening law. Not only concave softening behavior but also high-order cohesive traction, both being limitations of the previous works, can be properly dealt with. The global fracture responses are insensitive not only to the phase-field length scale but also to the traction order parameter, though the crack bandwidth might be affected by both. Despite the loss of variational consistency in general cases, the resulting μPF-CZM is still thermodynamically consistent. Moreover, the existing numerical implementation can be adopted straightforwardly with minor modifications. Representative numerical examples are presented to validate the proposed μPF-CZM and to demonstrate its capabilities in capturing brittle and cohesive fracture with general softening behavior. The insensitivity to both the phase-field length scale and the traction order parameter is also sufficiently verified.

Fracture behavior of thermal mismatch in functionally graded materials using phase-field modeling

A phase-field modeling is presented to study the complex thermal mismatch fracture behavior in functionally graded materials. The fracture resistance in functionally graded materials is homogenized by the rule of mixtures of the energy release rate. For validation, the bi-material plates and rectangular plates coated with functionally graded materials under thermal load are examined and the obtained results are compared with existing theoretical and experimental data. In the bi-material plates, the model captures commonly experimentally observed crack patterns such as the perpendicular crack, benching, and the formation of two pairs of parallel cracks. Additionally, a detailed comparison of the correlation between crack depth and film thickness is conducted among theoretical, experimental, and phase-field model data. Notably, the results of the phase-field model exhibit good agreement with experiments compared to the theoretical model. Subsequently, the model is applied to examine the thermal mismatch fracture behavior in plates of different shapes. The analysis emphasizes three key aspects: (i) examining the damage field to understand crack initiation and propagation, (ii) analyzing the major principal stress field in the coatings to identify fracture-critical regions, and (iii) tracking the evolution of elastic/fracture/total energies to investigate the fracture/deformation behavior. This analysis enhances a fundamental understanding of thermal fracture mechanisms in functionally graded materials and provides a foundation for optimizing materials and structures in various engineering applications.

A coupled crystal inelasticity-phase field model for crack growth in polycrystalline nitinol microstructures

Abstract This paper introduces a comprehensive computational framework, comprising a finite deformation crystal inelasticity constitutive model and phase field model, for modeling crack growth in superelastic nitinol polycrystalline microstructures. The crystal inelasticity model represents crystal stretching and lattice rotation from elastic mechanisms, as well as local inelastic deformation due to austenite-martensite phase transformation. The phase field formulation decomposes the Helmholtz free energy density into stored elastic energy, phase transformation energy, and crack surface energy components. The elastic energy accounts for tension-compression asymmetry with the formation of the crack through a spectral decomposition. Kinetic Monte Carlo simulations generate equilibrium area fractions of different surface orientations, which serve as weights for the surface energy. An adaptive wavelet-enhanced hierarchical finite element (FE) model is introduced to alleviate high computational overhead in phase field crack simulations. Simulations with the coupled inelasticity phase field model are conducted under various loading conditions including Mode-I tension, a quasi-static Kalthoff experiment, and cyclic loading of polycrystalline microstructures. Crack propagation is effectively predicted by this model, providing valuable insights into the material mechanical behavior with growing cracks.

Mixed-mode thermo-mechanical fracture: An adaptive multi-patch isogeometric phase-field cohesive zone model

This work presents an adaptive phase-field cohesive zone model (PF-CZM) for simulating mixed-mode crack nucleation and growth in isotropic rock-like materials subjected to thermo-mechanical interactions. The proposed approach combines an adaptive multi-patch isogeometric analysis (MP-IGA) and length-scale insensitive PF-CZM. The formulation captures the distinct critical energy release rates for Mode-I and Mode-II fractures, which is crucial for predicting mixed-mode thermo-mechanical fracture behavior in isotropic rock-like materials. The PF-CZM governing equations are solved with isogeometric analysis based on locally refined non-uniform rational B-splines (LR NURBS), and the complex structural geometry is exactly described with multiple LR NURBS patches. The field variables, such as displacement, phase-field, and temperature at the interface of adjacent patches, are coupled using Nitsche’s method. To enhance the computational efficiency while maintaining accuracy, a refinement-correction adaptive scheme combined with the structured mesh refinement strategy is developed. The proposed framework is validated against recent numerical and experimental results in the literature, particularly in the context of capturing complex behavior of mixed-mode crack propagation in isotropic rock-like materials subjected to thermo-mechanical loading.

Diffuse interface modeling of non‐isothermal Stokes‐Darcy flow with immersed transmissibility conditions

AbstractThe coupling between free and porous medium flows has received significant attention since it plays an important role in a wide range of problems from fluid‐soil interactions to biofluid dynamics. However, modeling this coupled process remains a difficult task as it often involves a domain decomposition algorithm in conjunction with a special treatment at the interface. The problem can become more challenging under non‐isothermal conditions because it requires the iterative procedure at every time step to simultaneously meet the transient mass continuity, force equilibrium, and energy balance for the entire system. This article presents a diffuse interface framework for modeling non‐isothermal Stokes‐Darcy flow and the corresponding finite element formulation that bypasses the need for explicitly splitting the domain into two, which enables the unified treatment for distinct regions with different hydrothermal flow regimes. To achieve this goal, we employ the Allen‐Cahn type phase field model to generate the diffuse geometry, where the solution field can be seen as a regularized approximation of the Heaviside indicator function, allowing us to transfer the interface conditions into a set of immersed boundary conditions. Our formulation suggests that the isothermal operator splitting strategy can be adopted without compromising accuracy if the heat and mass transfer processes are decoupled by assuming that the density and viscosity of the phase constituents are independent to the temperature. Numerical examples are also introduced to verify the implementation and to demonstrate the model capacity.

Phase field modeling for composite material failure

AbstractA computational framework of anisotropic phase field model is used to investigate the effects of effective critical energy release rate, interface property, and hole shape on the failure of composite material in this paper. This model is implemented in ABAQUS using a user‐defined element (UEL). First of all, the model is validated through a benchmark model. Second, the effective critical energy release rate is introduced, which can improve the accuracy of the simulation result. Then, the influence of strong and weak interfaces on the failure of composite material is studied. It can be found that the interface properties can change the evolution mode of crack and the final load–displacement response. Finally, the effect of hole shapes on the failure of the composite material is also explored, which shows that the bearing capacity of model can be improved by changing the shape of the hole.

Study of the dynamic impact spalling of ductile materials based on Gurson-type phase-field model

The formation of void damage and spalling failure in ductile metallic materials under strong impact is a well-established phenomenon. In aerospace and defense technology engineering design, understanding the spalling failure process and related mechanisms is of utmost importance. This paper develops an explicit Gurson-type phase-field model that can simulate the void evolution and spalling damage of three-dimensional ductile metallic materials under high-velocity impacts based on the study of Aldakheel et al.. The model incorporates the Gurson-type void evolution equation and the phase-field approach while taking into account the pressure-dependent bulk modulus and inertia effects. This model is used to study the main processes and mechanisms of impacted layer cracking of metals in different dimensions. Meanwhile based on the study of complex spallation cracking processes in metals in two and three dimensions, observing and proposing the formation mechanism of complex spallation cracking modes in materials due to lateral and edge (base angle) rarefied effects.

A phase-field gradient-based energy split for the modeling of brittle fracture under load reversal

In the phase-field modeling of fracture, the search for a physically reasonable and computationally feasible criterion to split the elastic energy density into fractions that may or may not contribute to crack propagation has been the subject of many recent studies. Within this context, we propose an energy split – or energy decomposition – aimed at accurately representing the evolution of a crack under load reversal. To this purpose, two key assumptions are made. First, the damage gradient direction is interpreted as being representative of the normal-to-crack direction, as already assumed in previous works in the literature. The second assumption consists of considering the sign of the projection of the stress tensor onto the damage gradient direction at a point as an indicator of whether this point should behave as an opening or as a closing crack. We associate the latter case (crack closing) to both (a) a complete recovery of elastic energy density of the intact material (i.e., perfectly rough crack surfaces) and (b) a zero crack driving force at that point. The first case (crack opening) is treated classically as a damageable material point at which damage can increase. The implementation of the proposed approach turns out to be remarkably simple and computationally robust. For the evaluation of the displacements and damage gradients at nodes, the classical Z2 technique is used, and a new effective and computationally convenient iterative strategy is implemented to guarantee convergence of the staggered scheme. Four examples are presented in order to assess the suitability of the present model by using both AT1 and AT2 regularization models. Results show the desired effect of limiting crack propagation to prevailing tensile states, as well as of recovering the initial intact stiffness upon load reversal, even when two of the most common energy splits fail.

Dynamical phase-field model of cavity electromagnonic systems

Cavity electromagnonic system, which simultaneously consists of cavities for photons, magnons (quanta of spin waves), and acoustic phonons, provides an exciting platform to achieve coherent energy transduction among different physical systems down to single quantum level. Here we report a dynamical phase-field model that allows simulating the coupled dynamics of the electromagnetic waves, magnetization, and strain in 3D multiphase systems. As examples of application, we computationally demonstrate the excitation of hybrid magnon-photon modes (magnon polaritons), Floquet-induced magnonic Aulter-Townes splitting, dynamical energy exchange (Rabi oscillation) and relative phase control (Ramsey interference) between the two magnon polariton modes. The simulation results are consistent with analytical calculations based on Floquet Hamiltonian theory. Simulations are also performed to design a cavity electro-magno-mechanical system that enables the triple phonon-magnon-photon resonance, where the resonant excitation of a chiral, fundamental (n = 1) transverse acoustic phonon mode by magnon polaritons is demonstrated. With the capability to predict coupling strength, dissipation rates, and temporal evolution of photon/magnon/phonon mode profiles using fundamental materials parameters as the inputs, the present dynamical phase-field model represents a valuable computational tool to guide the fabrication of the cavity electromagnonic system and the design of operating conditions for applications in quantum sensing, transduction, and communication.

Optimal Control of Material Microstructures

Abstract In this paper, we consider the optimal control of material microstructures. Such material microstructures are modeled by the so-called phase-field model. We study the underlying physical structure of the model and propose a data-based approach for its optimal control, along with a comparison to the control using a state-of-the-art reinforcement learning (RL) algorithm. Simulation results show the feasibility of optimally controlling such microstructures to attain desired material properties and complex target microstructures.