Phase-field Parameter Research Articles

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.

A phase field method for predicting hydrogen-induced cracking on pipelines

An accurate determination of the threshold conditions to initiate cracks on aged hydrogen pipelines is paramount for ensuring energy transport safety. In this work, a finite element-based phase field method was developed to assess the crack initiation on dented pipelines while considering the hydrogen (H) impact. Theoretical and multi-physics numerical formulas were derived for prediction of the elastic-plastic fracture behavior of H-contained steel. A critical phase field parameter, ϕ=0.69, is defined for predicting crack initiation at the dent on pipelines. The presence of H within the steel decreases the threshold dent depth for initiating H-induced cracks. When the initial H concentration increases from 0 to 0.5 wppm, the maximum dent depth for crack initiation reduces from 17.5 mm to 10.7 mm. The maximum dent depth required for crack initiation reduces from 17.5 mm to 7.8 mm when an internal pressure of 8 MPa is applied on the steel pipe. The site with the maximum phase field parameter changes during indentation, implying that the location initiating cracks depends on the dent dimension. The existing criteria in ASME B31.12 standard are not applicable for predicting H-induced crack initiation on dented pipelines. This study proposes a new method to predict hydrogen-induced cracking on aged pipelines when transporting hydrogen.

Microstructure-chemomechanics relations of polycrystalline cathodes in solid-state batteries

Lithium-nickel-manganese-cobalt-oxides (NMC) embedded in solid-electrolytes are being extensively applied as composite cathodes to match the high energy density of metallic anodes. During charge/discharge, the cathode composite often degrades through the evolution of micro-cracks within the grains, along the grain boundaries, and delamination at the particle-electrolyte interface. Experimental evidence has shown that regulating the morphology of grains and their crystallographic orientations is an effective way to relieve the volume-expansion-induced stresses and cracks, consequently stabilizing the electrochemical performance of the electrode. However, the interplay among the crystal orientation, grain morphology, and chemo-mechanical behavior has not been holistically studied. In that context, a thermodynamically consistent computational framework is developed to understand the role of microstructural modulation on the chemo-mechanical interactions of a polycrystalline NMC secondary particle embedded in a sulfide-based solid electrolyte. A phase-field fracture variable is employed to consider the initiation and propagation of cracks. A set of diffused phase-field parameters is adopted to define the transition of chemo-mechanical properties between the grains, grain boundaries, electrolyte, and particle-electrolyte interfaces. This modeling framework is implemented in the open-source finite element package MOOSE to solve three state variables: concentration, displacement, and phase-field damage parameter. A systematic parametric study is performed to explore the effects of aspect ratio, the crystal orientation of grains, and the interfacial fracture energy through the chemo-mechanical analysis of the composite electrode. The findings of this study offer predictive insights for designing solid-state batteries that provide stable performance with reduced fracture evolution.

Simulation of crack propagation in piping components using phase field approach

Understanding the fracture behaviour in pipelines transporting steam and other resources is vital for its longevity and structural integrity. The prediction of crack initiation and propagation, involving crack branching, and the nucleation of new cracks, etc. in fracture processes are challenging tasks. The phase field method (PFM), based on variational formulation, emerged as a competitive mathematical tool to address fracture mechanics problems. PFM does not require the presence of predefined cracks, handles the crack initiation and propagation within the same system. This feature provides an added advantage to PFM and overcomes the limitations of Griffith’s approach-based linear elastic fracture mechanics (LEFM). In the present work, simulation of the crack propagation in the piping components by employing phase field methodologies has been carried out. Two parameters namely; a phase field parameter (ϕ) that varies between 0 and 1 differentiating between broken and intact material, and a length scale parameter (l_0) that approximates the sharp crack into the diffused crack with exponential distribution are introduced in the mathematical modeling. Numerical modeling is solved using the staggered algorithm in three layered scheme defining the displacement field and phase field within a 3-D finite element framework and implemented in Abaqus via user-defined element (UEL) subroutine codes. PFM simulates the crack path and predicts the load-displacement response, which is validated with benchmark problems, and extended for piping components under monotonic loading. The proposed study accurately captures the crack growth behaviour as observed in experiments on SA312 Type 304LN stainless steel pipes and hence, demonstrated the robustness of the proposed formulation.

Numerical modeling of stress corrosion cracking in steel structures with phase field method

This study uses the phase field method to present a coupled mechano-electro-chemical formulation for predicting stress corrosion cracking (SCC) phenomena in steel structures. SCC constitutes an intricate damage process stemming from the interplay between mechanical loading and corrosion within an aggressive environment. The formulation presented herein introduces a phase-field parameter that aggregates the mechanical and electrochemical influences. To achieve this goal, the internal energies governing the SCC phenomenon are separated into elastic-damage strain energy, interfacial reaction energy, and energy resulting from changes in corrosion ion concentration. The Allen-Cahn equation is modified to include all energy contributions and calculate the phase-field parameter. Furthermore, a specific interfacial kinetic coefficient is introduced to the mechanical energy to consider corrosion current effects on mechanical properties. The Cahn-Hilliard equation is applied to model the corrosion ion concentration in the domain, and the mechanical state of the body is obtained by solving the equilibrium equations. Several numerical examples are presented to indicate the robustness and accuracy of the proposed formulation. In this respect, the method is applied to predict crack propagation resulting from SCC in two practical engineering problems, yielding promising results.

Phase field modeling of fatigue crack growth retardation under single cycle overloads

A zone-based crack retardation model that responds to single cycle overloads is applied to a phase field fatigue framework. Fatigue crack growth is incorporated through the degradation of fracture toughness in the phase field fracture model where a representative loading strategy is followed instead of explicit cyclic loading. The model is demonstrated to capture Paris-Erdogan law type behavior. Crack growth retardation following an overload cycle is simulated through a zone ahead of the crack tip taking inspiration from existing plastic zone-based retardation models. The retardation zone is described through a strain energy density limit, with the overload ratio controlling the extent to which the fatigue damage accumulation rate is slowed inside the retardation zone. The model is implemented utilizing user subroutines in Abaqus with coupled-temperature displacement elements where temperature acts as a stand in for the phase field parameter. The model is found capable of reproducing experimental levels of fatigue life gains and reduction in crack growth rate for compact tension and center-cracked tension panel specimens when various overload levels are applied. Furthermore, the model is demonstrated to be able to capture complex crack paths with great accuracy.

Dynamic and adaptive mesh-based graph neural network framework for simulating displacement and crack fields in phase field models

Fracture is one of the main causes of failure in engineering structures. Phase field methods coupled with adaptive mesh refinement (AMR) techniques have been widely used to model crack propagation due to their ease of implementation and scalability. However, phase field methods can still be computationally demanding making them unfeasible for high-throughput design applications. Machine learning (ML) models such as Graph Neural Networks (GNNs) have shown their ability to emulate complex dynamic problems with speed-ups orders of magnitude faster compared to high-fidelity simulators. In this work, we present a dynamic mesh-based GNN framework for emulating phase field simulations of single-edge crack propagation with AMR for different crack configurations. The developed framework – ADAPTive mesh-based graph neural network (ADAPT-GNN) – exploits the benefits of both ML methods and AMR by describing the graph representation at each time step as the refined mesh itself. Using ADAPT-GNN, we predict the evolution of displacement fields and scalar damage field (or phase field) with good accuracy compared to the conventional phase-field fracture model. We also compute crack stress fields with good accuracy using the predicted displacements and phase field parameter.

Controls Stack Pressure Induced Fracture Using Microstructural Modulation in Solid-State Batteries

All-solid-state batteries (ASSBs) provide higher energy densities and safer alternatives to Li-ion batteries by incorporating Li-metal anode and inflammable solid electrolytes. To match the increased capacity of metallic anodes, ASSBs require high-energy-density cathode materials such as LiNixCoyMn1-x-yO2 (also known as NMC cathode) blended with a solid-state electrolyte (SSE). However, a key challenge is resolving the interfacial incompatibility between the active particulate material and the solid-state electrolyte within these composite cathodes. To obtain intimate contact between SSE and the active material, an external load (or stack pressure) needs to be applied during the processing and service life of cell. Nevertheless, such densification of cathode composites can lead to grain boundary fracture and/or complete particle disintegration. Therefore, the present work utilizes a microstructural modulation procedure to alleviate stack pressure-induced fracture in a polycrystalline cathode. Accordingly, a thermodynamically consistent computational framework is developed to understand the interplay between the stack pressure, microstructural modulation, and fracture behavior for polycrystalline NMC secondary particles embedded in a sulfide-based solid electrolyte. A phase-field fracture variable is employed to consider the initiation and propagation of cracks in the active material and SSE. This modeling framework is implemented in the open-source finite element package (MOOSE) to solve three state variables: concentration, displacement, and the phase-field damage parameter. A systematic parametric study is performed to explore the effects of stack pressure, aspect ratio, and the crystal orientation of grains on the chemo-mechanical performance of the composite electrode. We also quantitatively validate the numerical model with experimental investigation using state-of-the-art Nano-CT (Compact tomography) technique. The findings of this study offer predictive insights for designing solid-state batteries with reduced fracture evolution and stable performance. Figure 1

Phase-field modeling of stochastic fracture in heterogeneous quasi-brittle solids

Owing to the random nature of heterogeneity, damage and fracture behavior of quasi-brittle materials exhibits a considerable degree of uncertainty. Computational modeling of stochastic fracture in quasi-brittle materials has become an indispensable tool for analysis and design of engineering structures. To this end, we present in this paper a computational framework to capture probabilistic fracture in heterogeneous quasi-brittle solids by combining the random field theory and the phase-field cohesive zone model (PF-CZM). The spatial variation of the material strength and fracture energy is represented by a cross-correlated bivariate random field generated by the Karhunen–Loève expansion. The recently proposed PF-CZM is employed to simulate the stochastic crack nucleation and propagation in quasi-brittle solids. The objectivity of the Monte-Carlo simulation is achieved by imposing a specific condition on the phase-field length scale parameter and the correlation length of the random field. In particular, upon this condition the width of the fracture process zone (FPZ) is considerably smaller than the correlation length of the random field such that the material inside the FPZ does not exhibit significant spatial variations of mechanical properties. As the fracture energy is intrinsically incorporated in the PF-CZM, it is unnecessary in this case to explicitly consider the FPZ width. The resulting probabilistic PF-CZM together is applied to the Monte-Carlo simulations of fracture in concrete structures of different geometries. It is shown that the stochastic simulation results are insensitive to both the phase-field length scale parameter and the finite element mesh discretization as in the previous deterministic analyses. Enhanced with the specific condition on the involved characteristic lengths, the PF-CZM provides a viable tool for stochastic simulations of damage and fracture in quasi-brittle structures.

Phase field modeling and computation of vesicle growth or shrinkage.

We present a phase field model for vesicle growth or shrinkage induced by an osmotic pressure due to a chemical potential gradient. The model consists of an Allen-Cahn equation describing the evolution of the phase field parameter that describes the shape of the vesicle and a Cahn-Hilliard-type equation describing the evolution of the ionic fluid. We establish conditions for vesicle growth or shrinkage via a common tangent construction using free energy curves. During the membrane deformation, the model ensures total mass conservation of the ionic fluid, and we weakly enforce a surface area constraint of the vesicle. We develop a stable numerical scheme and an efficient nonlinear multigrid solver to evolve the phase and concentration fields, and we use this to evolve the fields to near equilibrium for 2D vesicles. Convergence tests confirm an [Formula: see text] accuracy for our scheme and near-optimal convergence for our multigrid solver. Numerical results reveal that the diffuse interface model captures the main features of cell shape dynamics: for a growing vesicle, there exist circle-like equilibrium shapes if the concentration difference across the membrane and the initial osmotic pressure are large enough; while for a shrinking vesicle, there exists a rich collection of finger-like equilibrium morphologies.

An interface preserving and residual-based adaptivity for phase-field modeling of fully Eulerian fluid-structure interaction

In this paper, we present a novel interface preserving and residual-based adaptivity procedure for the phase-field modeling of a fully Eulerian fluid-structure interaction. The proposed adaptive fully Eulerian procedure involves a fixed background unstructured finite element mesh on which the fluid-structure boundary is treated implicitly via a diffused interface description. We employ the stabilized finite element formulation and a partitioned iterative procedure to solve the coupled system of the Allen-Cahn phase-field equation with the unified momentum equation comprising solid and fluid dynamics. To evaluate the solid stresses, the left Cauchy-Green deformation tensor is convected at each time step to trace the evolution of the solid strain in the Eulerian reference frame. We utilize a residual based error indicator and the newest vertex bisection algorithm for the adaptive refinement/coarsening of the unstructured mesh. Through the adaptive finite element method, we aim to preserve the profile of the diffused interface in the presence of convective distortion. We propose an early stopping criterion for the adaptivity of the interface dynamics by combining the effects of the phase-field interface thickness parameter and the mesh resolution. In addition, we control the size of the refined/coarsened elements by introducing a characteristic length scale governed by a weighted harmonic mean of the desired bulk and the interface mesh resolutions. The proposed stopping criterion and the characteristic length scale provide significant reductions in the computational cost while simultaneously ensuring convergence properties of the coupled fluid-structure system. We perform a detailed convergence and accuracy analysis via two benchmark problems namely, the pure solid system and a coupled fluid-solid system with an interface in a rectangular domain. We further provide a comparative study between feature- and residual based adaptive strategies. We systematically assess the performance of the adaptive procedure in terms of conservation properties and computational efficiency. Finally, we demonstrate our fully Eulerian adaptive solver to simulate the contact and bouncing phenomenon between an elastic solid and a rigid wall.

Active learning sensitivity analysis of γ'(L12) precipitate morphology of ternary co-based superalloys

To better understand the equilibrium γ′(L12) precipitate morphology in Co-based superalloys, a phase field modeling sensitivity analysis is conducted to examine how four phase-field parameters [initial Co concentration (c0), double-well barrier height (ω), gradient energy density coefficient (κ), and lattice misfit strain (ϵmisfit)] influence the γ′(L12) precipitate size and morphology. Gaussian Process Regression (GPR) models are used to fit the sample points and to generate surrogate models for both precipitate size and morphology. In an Active Learning approach, a Bayesian Optimization algorithm is coupled with the GPR models to suggest new sample points to calculate and efficiently update the models based on a reduction of uncertainty. The algorithm has a user-defined objective, which controls the balance between exploration and exploitation for new suggested points. Our methodology provides a qualitative and quantitative relationship between the γ′(L12) precipitate size and morphology and the four phase-field parameters, and concludes that the most sensitive phase-field parameter for precipitate size and morphology is the initial Co concentration (c0) and the double-well barrier height (ω), respectively. We note that the GPR model for precipitate morphology required adding a noise tolerance in order to avoid overfitting due to irregularities in some of the simulated equilibrium γ′(L12) precipitate morphology.

Thermo4PFM: Facilitating Phase-field simulations of alloys with thermodynamic driving forces

Phase-field modeling is a popular front-tracking approach used to model solidification. Its time-evolution equations are often coupled to alloy composition and/or thermal diffusion in high-resolution multiphysics approaches. Materials thermodynamic properties tabulated in CALPHAD databases can be used for phase-field modeling to parameterize bulk energies of alloys. In addition, they can be naturally integrated into models such as the Kim-Kim-Suzuki (KKS) model where driving forces depend on the differences between chemical potentials of co-existing phases. In that case, a small system of coupled nonlinear equations needs to be solved at every point in space where the phase-field order parameter is to be updated and evolved in time. We present Thermo4PFM, a solver for the KKS equations for binary and ternary alloys, with two or three phases, and parameterized with CALPHAD models. Thermo4PFM is open source, written in C++, and can take advantage of Graphics Processing Units (GPU) accelerators. Using OpenMP offload capabilities for C++ classes, an excellent performance is demonstrated on GPU using the LLVM compiler. CALPHAD data is read from simple JSON files using an open source parser from the boost library. Program summaryProgram Title: Thermo4PFMCPC Library link to program files:https://doi.org/10.17632/8j3ntp5c7k.1Developer's repository link:https://github.com/ORNL/Thermo4PFMLicensing provisions: BSD 3-clauseProgramming language: C++/OpenMPNature of problem: Accurate modeling of solidification in metallic alloys requires thermodynamic data associated with possible material phases. That data can be used in phase-field modeling (PFM) to evaluate the driving force responsible for phase changes and solidification front motion. Integrating that data into PFM requires the solution of a small system of coupled nonlinear differential equations — the Kim-Kim-Suzuki equations — that needs to be solved at every point of a discretization mesh.Solution method: Thermo4PFM implements a Newton-based solver for the Kim-Kim-Suzuki equations for a few special cases (binary and ternary alloys with two or three phases) for CALPHAD-based models of the bulk energy of the phases. The software is written in C++ and uses the Curiously Recurring Template Pattern and OpenMP offload capabilities to take advantage of GPU accelerators, when available, on modern High Performance Computing resources. It also uses the boost Property Tree library to parse input CALPHAD data.

Three dimensional finite element computation of the non-isothermal polymer filling process by the phase field model

In this paper, we numerically study the three dimensional non-isothermal non-Newtonian filling process using the phase field method. The Cahn-Hilliard equation is firstly attempted to capture the moving melt front between the Newtonian and non-Newtonian fluid. The governing equations of the flow field are able to be written in a unified form on the basis of phase field parameter. The unified Navier-Stokes equations are decoupled via the splitting scheme and then the traditional FEM and SUPG method are employed to solve the suitable equations. The Cross-WLF model is utilized to describe the variation of the viscosity of the polymer melt in the non-isothermal condition. We take the thin wall rectangular cavity as example and compare with the experimental data to illustrate the convergence, robustness and accuracy of numerical algorithm. In addition, we consider the thin wall cavity with a cylindrical obstacle and analyze the influences of the inlet velocity and the size of the injection gate on the filling process. The numerical results agree well with the experimental data and also exhibit good mass conservation property.

A Posteriori Estimator for the Adaptive Solution of a Quasi-Static Fracture Phase-Field Model with Irreversibility Constraints

Within this article, we develop a residual type a posteriori error estimator for a time discrete quasi-static phase-field fracture model. Particular emphasize is given to the robustness of the error estimator for the variational inequality governing the phase-field evolution with respect to the phase-field regularization parameter $\epsilon$. The article concludes with numerical examples highlighting the performance of the proposed a posteriori error estimators on three standard test cases; the single edge notched tension and shear test as well as the L-shaped panel test.

Upscaling of a Cahn–Hilliard Navier–Stokes model with precipitation and dissolution in a thin strip

We consider a phase-field model for the incompressible flow of two immiscible fluids. This model extends widespread models for two fluid phases by including a third, solid phase, which can evolve due to e.g. precipitation and dissolution. We consider a simple, two-dimensional geometry of a thin strip, which can still be seen as the representation of a single pore throat in a porous medium. Under moderate assumptions on the Péclet number and the capillary number, we investigate the limit case when the ratio between the width and the length of the strip goes to zero. In this way, and employing transversal averaging, we derive an upscaled model. The result is a multi-scale model consisting of the upscaled equations for the total flux and the ion transport, while the phase-field equation has to be solved in cell problems at the pore scale to determine the position of interfaces. We also investigate the sharp-interface limit of the multi-scale model, in which the phase-field parameter approaches 0. The resulting sharp-interface model consists only of Darcy-scale equations, as the cell problems can be solved explicitly. Notably, we find asymptotic consistency, that is, the upscaling process and the sharp-interface limit commute. We use numerical results to investigate the validity of the upscaling when discontinuities are formed in the upscaled model.

A phase field model for partially saturated geomaterials describing fluid–fluid displacements, Part II: Stability analysis and two-dimensional simulations

Flows involving immiscible displacement of one fluid by another in a porous media are known to destabilize and form fluid fingering. When the non-wetting fluid is a highly mobile gas (air) and the wetting fluid is an in-compressible liquid (water) the classical macroscopic theory is unable to describe the fingered flow. In Part I of this study we have introduced a model that interprets the mixture of wetting and non-wetting fluids within the pore space as a single saturating non-uniform pore fluid characterized by a phase field parameter, which is considered to be the saturation degree of the wetting fluid. In the current study we present a linear stability analysis of its solutions which describe both imbibition and drainage. The analysis sheds light on the sensitivity of the flow stability on injection flux, imposed pressure gradient and initial saturation degree. Two-dimensional numerical simulation results are as well presented which verify the stability analysis and reveal the rich structure of the fluid fingering realized by this model. While these results are found to be in qualitative agreement with experimental observations, they also warrant further experimentation to explore the additional features predicted by the model.

A phase field model for partially saturated geomaterials describing fluid–fluid displacements. Part I: The model and one-dimensional analysis

A model describing immiscible fluid–fluid displacements in partially saturated porous media is presented. This is based on a phase field approach that interprets the mixture of wetting (liquid water) and non-wetting (air) fluids within the pore space as a single saturating non-uniform pore fluid characterized by a phase field parameter, which is considered to be the saturation degree of the wetting fluid. While the standard retention curve provides for the retention properties of the pore walls, a Cahn–Hilliard like double-well energy is employed to describe the possible co-existence of the immiscible fluid phases. An enhanced description of the macroscopic surface tension between the fluid phases is obtained naturally within the phase field framework due to a regularization that depends on the spatial gradient of the water content. A generalized Darcy’s law is used to describe dissipation due to fluid flow driven by the gradient of a generalized chemical potential. Thus, in the context of soil hydrology this model is interpreted as an extension to the classical Richards equation governing the spatio-temporal evolution of the phase field parameter. Employing a convex–concave flux function it is shown, using one-dimensional analysis, that both imbibition and drainage fronts can be modeled in this phase field framework. The non-monotonicities observed in the resolved solutions are explained using a combination of asymptotic matching techniques and dynamical systems analysis.

Adaptive wavelet-enhanced cohesive zone phase-field FE model for crack evolution in piezoelectric composites

A novel finite deformation FE model is developed in this paper for coupled electro-mechanical response with crack evolution in heterogeneous piezoelectric composite microstructures. Fracture in these composites is driven by complex mechanical and electrical mechanisms leading to a set of strongly coupled, nonlinear governing equations. Crack propagation and interfacial decohesion in the presence of electro-mechanical fields are modeled by a unified phase-field model integrated with cohesive traction–separation laws, derived from a cohesive potential function at material interfaces. To incorporate different electrical crack face conditions, degradation functions are employed in terms of the crack phase-field order parameter. An adaptive wavelet-enhanced hierarchical framework is developed to improve computational efficiency of the coupled electromechanical-phase-field FE solver based on a phase-field error indicator. Numerical examples representing different failure mechanisms in heterogeneous piezoelectric materials and piezo-composite microstructures are solved to demonstrate the efficacy of the model.

A decoupled, stable, and linear FEM for a phase-field model of variable density two-phase incompressible surface flow

The paper considers a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids. The model allows for a non-linear dependence of the fluid density on the phase-field order parameter. Driven by applications in biomembrane studies, the model is written for tangential flows of fluids constrained to a surface and consists of (surface) Navier–Stokes–Cahn–Hilliard type equations. We apply an unfitted finite element method to discretize the system and introduce a fully discrete time-stepping scheme with the following properties: (i) the scheme decouples the fluid and phase-field equation solvers at each time step, (ii) the resulting two algebraic systems are linear, and (iii) the numerical solution satisfies the same stability bound as the solution of the original system under some restrictions on the discretization parameters. Numerical examples are provided to demonstrate the stability, accuracy, and overall efficiency of the approach. Our computational study of several two-phase surface flows reveals some interesting dependencies of flow statistics on the geometry.