Variational Phase Field Research Articles

Nucleation of Fracture: The First-Octant Evidence Against Classical Variational Phase-Field Models

Abstract As a companion work to [1], this article presents a series of simple formulae and explicit results that illustrate and highlight why classical variational phase-field models cannot possibly predict fracture nucleation in elastic brittle materials. The focus is on “tension-dominated” problems where all principal stresses are nonnegative, that is, problems taking place entirely within the first octant in the space of principal stresses.

Phase-field finite element modelling of creep crack growth in martensitic steels

Creep fracture presents a major concern for structural materials and critical components operating at elevated temperatures, thus requiring effective computational models. This study presents a phase-field framework for modelling creep crack growth and fracture behaviour of modern high-temperature materials such as Creep Strength Enhanced Ferritic (CSEF) martensitic steels. The model was formulated using the thermodynamic principles of the variational phase field theory of fracture and considering some physical aspects of creep fracture. Within the modelling framework, a dissipation potential dependent on creep damage is introduced to capture the effect of creep cavitation on the fracture energy and the creep crack growth resistance of the solid in a phenomenological manner. An elastoplastic power-law creep model is coupled to the phase-field formulations to account for the non-linear deformation processes ahead of the crack tip due to inelastic deformations, as well as their contribution to fracture at high temperatures. The capability of the proposed model is assessed against experimental data from compact tension (CT) creep tests conducted on P91 and P92 steels. Good agreement was obtained between the FE-predicted creep crack growth behaviour and the experimental measurements, showcasing the model’s feasibility. Further, numerical experiments were conducted using the proposed model to elucidate some key aspects influencing the fracture behaviour of martensitic steels. The proposed computational framework not only demonstrated good capability but was also able to offer improved mechanistic insights into the influence of material tendency to develop creep cavities on crack growth behaviour. This work contributes valuable insights into understanding the fracture process of CSEF steels at elevated temperatures and further demystifies the role of creep ductility.

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

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

Verification of strain energy splits of phase field fracture model using Westergaard’s problem under mixed-mode loading

One challenge in verifying strain energy splits in phase field fracture models is the lack of reference problems with exact solutions in mixed-mode loading that could serve as benchmark problems. In this work, different strain energy splits used in the variational phase field model have been applied to the classical Westergaard’s problem. In contrast to other benchmark problems used in the literature, the proposed problem not only includes the singular term as a boundary condition but also implicitly considers all terms. To the authors’ knowledge, this is the first time in the literature that this reference problem has been used to rigorously verify the accuracy of different energy splits of the phase field model. Several critical factors such as the pre-crack definition, mesh size, mixed mode ratio and the length scale parameter have been analysed. A good correlation between the numerical estimations and the analytical predictions was found for some pre-crack definition approaches under mode I. However, we demonstrated that the compatible regularisation length scale and mesh size are significantly lower under mode II loading (i.e., when the crack kinks) than under mode I.

Modeling of Hydro-mechanical Coupled Fracture Propagation in Quasi-brittle Rocks Using a Variational Phase-Field Method

Modeling of Hydro-mechanical Coupled Fracture Propagation in Quasi-brittle Rocks Using a Variational Phase-Field Method

Meso-scale size effects of material heterogeneities on crack propagation in brittle solids: Perspectives from phase-field simulations

Brittle solids are often toughened by adding a second-phase material. This practice often results in composites with material heterogeneities on the meso scale: large compared to the scale of the fracture process zone but small compared to that of the application. The specific configuration (both geometrical and mechanical) of this mesoscale heterogeneity is generally recognized as important in controlling crack propagation behavior and, subsequently, the (effective) toughness of the composite. Here, we systematically investigate how dynamic brittle fracture navigates through a linear array of mesoscale inclusions. Using a variational phase-field (PF) approach, we compute the apparent crack speed and fracture energy dissipation rate to compare crack propagation (and the resulting toughening) under Mode-I loading for various configurations of inclusions. We identify an interplay between the size of inclusion and that of the K-dominant zone in the presence of elastic heterogeneity: matching these two sizes gives rise to the best toughening outcome for a given area fraction of inclusions. We discuss mechanisms that rationalize this observation and the importance of the length scale parameter used in PF models in interpreting simulation results. Our work sheds physical insight into the interaction between size effects and material properties, thereby opening a venue for the rational design of functional (architected) composites for dynamic fracture applications.

A generally variational phase field model of fracture

Although the phase field method has been widely used in the field of fracture, various models exist, such as the second-order and fourth-order phase field model in the field of fracture mechanics, and the phase field evolution model in the field of physics, and it is not yet possible to establish these models under a unified framework. In this paper, a thermodynamically generalized framework of variational phase field model of fracture is presented, which can effectively unify different models under this framework. Based on this variational framework, not only the current second-order and fourth-order phase field model can be unified, but also new second-order, fourth-order, sixth-order, and eighth-order phase field model can be obtained from this framework. In addition, a method is proposed to convert the lower-order model into a higher-order model, which can convert a phase field distribution function of the lower-order model into a high-order phase field distribution function, and then obtain a higher-order phase field model. Finally, the new model is verified by some representative examples, and the second-order and fourth-order phase field models are compared, which shows the effectiveness of building a new phase field model from the variational framework.

A framework to model the hydraulic fracturing with thermo-hydro-mechanical coupling based on the variational phase-field approach

In this research, a numerical framework for hydraulic fracturing has been formulated, incorporating thermo-hydro-mechanical (THM) coupled effects. While previous studies have reported various hydraulic fracturing models based on the phase-field method, a THM coupling scheme grounded in the variational phase-field approach remains unexplored. The THM coupling is of paramount importance for understanding underground fracture propagation. This study’s principal innovation lies in its extension of the variational phase field model, introducing it to a thermoporoelastic medium, thereby integrating fluid flow and heat transfer elements in fractured materials. In our advanced model, the fluid flow and heat transfer in the matrix and fracture are modeled independently. Subsequently, employing a set of reasonable assumptions, we introduce unified flow and heat transfer equations derived from phase-field calculus. A pivotal contribution of our work is the development of an iterative solution algorithm based on the fixed stress method, pivotal for tackling the THM coupled system. The proposed equations are discretized employing the FEM. Additionally, to address the numerical oscillations in advection-dominant issues that manifest in the heat transfer model, we have applied a generalized Streamline Upwind Petrov Galerkin method for 8-node elements. The accuracy of fracture propagation was assessed by comparing the numerical results of the evolution of fracture length, fracture width at the injection point, and the injection pressure under the K regime with the corresponding analytical solution. Furthermore, the process of heat transfer within a fractured domain, often referred to as the Barends problem, has been studied by proposed model. Additionally, some numerical experiments were conducted to gain insights into the intricate interplay between the thermal–hydraulic–mechanical system during hydraulic fracturing, thereby further emphasizing the significance of our advanced model.

A quantitative variational phase field framework

The finite solid–liquid interface width in phase field models results in non-equilibrium effects, including solute trapping. Prior phase field modeling has shown that this extra degree of freedom, when compared to sharp-interface models, results in solute trapping that is well captured when realistic parameters, such as interface width, are employed. However, increasing the interface width, which is desirable for computational reasons, leads to artificially enhanced trapping thus making it difficult to model departure from equilibrium quantitatively. In the present work, we develop a variational phase field model that guarantees a temporal decrease in the free energy with independent kinetic equations for the solid and liquid phases. Separate kinetic equations for the phase concentrations obviate the assumption of point wise equality of diffusion potentials, as is done in previous works. Non-equilibrium effects such as solute trapping, drag and interface kinetics can be introduced in a controlled manner in the present model. In addition, the model parameters can be tuned to obtain “experimentally-relevant” trapping while using significantly larger interface widths than prior efforts. A comparison with these other phase field models suggests that interface width of about three to twenty-five times larger than current best-in-class models can be employed depending upon the material system at hand leading to a speed-up by a factor of W(d+2), where W and d denote the interface width and spatial dimension, respectively. Finally the capacity to model non-equilibrium phenomena is demonstrated by simulating oscillatory instability leading to the formation of solute bands.

Approximation of multiphase mean curvature flows with arbitrary nonnegative mobilities

This paper is devoted to the robust approximation with a variational phase field approach of multiphase mean curvature flows with possibly highly contrasted mobilities. The case of harmonically additive mobilities has been addressed recently using a suitable metric to define the gradient flow of the phase field approximate energy. We generalize this approach to arbitrary nonnegative mobilities using a decomposition as sums of harmonically additive mobilities. We establish the consistency of the resulting method by analyzing the sharp interface limit of the flow: A formal expansion of the phase field shows that the method is of second order. We propose a simple numerical scheme to approximate the solutions to our new model. Finally, we present some numerical experiments in dimensions 2 and 3 that illustrate the interest and effectiveness of our approach, in particular for approximating flows in which the mobility of some phases is zero.

Approximation of surface diffusion flow: A second-order variational Cahn–Hilliard model with degenerate mobilities

This paper addresses the approximation of surface diffusion flow using Cahn–Hilliard-type models. We introduce and analyze a new variational phase field model that associates the classical Cahn–Hilliard energy with two degenerate mobilities and guarantees a second-order accuracy in the approximation of the sharp limit. We also introduce simple and efficient numerical schemes to approximate the solutions to three Cahn–Hilliard-type models for surface diffusion flow, including the one we propose, and we provide 2D and 3D numerical experiments that illustrate the advantages of our approach.

Vibration characteristics of cracked FG-GRC plates in thermal environments based on phase field theory and meshless method

Stiffness degradation resulting from matrix cracking has a great impact on the vibrational behaviors of composite laminates. However, the vibration frequencies of functionally graded graphene-reinforced composite (FG-GRC) plates with crack defects remain largely unexplored. In this paper, the variational phase field theory is implemented to investigate the vibration characteristics of cracked FG-GRC plates subjected to thermal conditions. The effective elastic constants depending on the temperature of FG-GRC plates are characterized by the extended Halpin-Tsai model. The first-order shear deformation theory (FSDT) is employed to describe the displacement fields of the cracked FG-GRC plates. The meshfree kernel particle method is engaged to discretize governing equations over the computational domain and then the obtained governing eigen-equations are calculated using the Ritz methodology. The numerical results illustrate the influences of crack length, crack number, the strength of foundations, graphene distribution, temperature variation and geometric values on the vibration fundamental frequencies of the FG-GRC plates. The present work not only provides an alternative and feasible approach for solving the interaction problems between cracking and vibration, but also makes significant advances to characterize the vibrational behaviors of cracked functionally graded materials for engineering applications.

An explicit representation of cracks in the variational phase field method for brittle fractures

The phase field method (PFM) by means of regularization fractures has become a promising and popular method for complex crack propagation, especially in geotechnical applications. Attractive as it is, a challenge exists for the PFM to obtain explicit representation of cracks and jump displacement field across cracks that are essential and crucial for some applications, e.g., hydraulic fracturing, leakage problems, and contact problems. In the present paper, an algorithm in which the values of phase field are regarded as the indicator to identify crack paths is proposed. Then physical patches are cut by the reconstructed crack paths to obtain explicit cracks and jump displacement field in the context of the numerical manifold method (NMM). The numerical examples investigate that various explicit cracks, including a straight crack, a curve crack, the branching cracks, and the merging cracks, predicted by the proposed method are in good agreement with the literature. Besides, the Sneddon’s example verifies that the discontinuous displacement field across explicit cracks is in line with the closed form solution. The proposed numerical model deserves a further investigation.

Numerical Discretization of Variational Phase Field Model for Phase Transitions in Ferroelectric Thin Films

Phase field methods have been widely used to study phase transitions and polarization switching in ferroelectric thin films. In this paper, we develop an efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes to demonstrate the effectiveness of the proposed method.

A Thermodynamically Consistent Variational Framework to Model Lithium Nucleation and Subsequent Plating Under Fast Charge Conditions

Based on US Department of Energy’s stated goal for extreme fast charging the electric vehicle batteries need to charge in 10-15 min, which requires drastic improvement upon existing state-of-the-art batteries. One of the primary challenges under fast charge conditions is the lithium plating on graphite anode, which severely degrades the performance and cycle life. Further, the plated lithium grows as dendrites often result in a catastrophic failure. A variational phase field formulation has been developed, which enables to initiate the nucleation of lithium particles based on the local plating overpotential conditions and further allowing to predict the lithium plating under fast charge conditions. In this work we demonstrate that at small charging current density, there is no nucleation and plating of metallic lithium, and no effect on the intercalation kinetics. But from intermediate to fast charging current densities, a stagnated growth from isolated lithium nuclei is observed followed by growth into large nuclei and a continuous growth of metallic lithium. The formulation sets the stage for predicting and evaluating the influence of various electrolyte compositions, graphite particle morphologies and electrode loadings on the nucleation and plating dynamics under fast charge conditions.

Modeling of Lithium Nucleation and Plating Kinetics Under Fast Charge Conditions

The onset of lithium nucleation and subsequent plating on the graphite particles is a challenging problem in the design of fast charging batteries. A variational phase field formulation has been developed to initiate the nucleation of lithium particles based on the local plating overpotential conditions and allowing to predict the lithium plating under fast charge conditions. The model demonstrates that (1) at small charging current density, there is no nucleation and plating of metallic lithium, and no effect on the intercalation kinetics; (2) at intermediate charging current densities, initially a stagnated growth from isolated lithium nuclei is observed followed by a continuous growth of metallic lithium; (3) at large charging current densities, a continuous growth of metallic lithium resulting from coalescence of small nuclei into large nuclei decreases the excess interfacial energy, which in turn slows down the intercalation kinetics. The formulation sets the stage for predicting and evaluating the influence of various electrolyte compositions, graphite particle morphologies and electrode loadings on the nucleation and plating dynamics under fast charge conditions.

Bifurcation criterion and the origin of limit crack velocity in dynamic brittle fracture

By employing a variational phase field model, we investigated the dynamic brittle fracture of poly (methyl methacrylate) (PMMA) through computer simulation. Without the explicit fracture criterion, current simulations not only successfully reproduced the crucial experimental observations on crack propagation, such as crack patterns, velocity evolutions and limit crack velocity but also discovered some new features not reported in the experimental study yet. Through quantifying the energy flux into the crack tip and fracture energy, we observed that the crack bifurcation of PMMA always occurs with the energy flux into the crack tip exceeding a critical value and proposed that crack bifurcation obeys an energy criterion. Based on this criterion, the long-standing challenge of limit crack velocity in experiments is successfully predicted by continuum theory, which unveils that crack bifurcation sets the upper limit for crack velocity. Combining the crack bifurcation criterion and continuum theory provides a rational explanation for complex path selection of cracks.

In-situ strength effects in long fibre reinforced composites: A micro-mechanical analysis using the phase field approach of fracture

Transverse intralaminar cracks in layers with perpendicular orientation referred to the main loading direction have a significant effect on the apparent ultimate strength of the corresponding composite laminate. This effect stems from the fact that such transverse cracks generally promote the occurrence of other failure mechanisms leading to the specimen collapse in subsequent stages of the loading process. With the aim of conducting a careful investigation regarding the onset and progression of transverse intralaminar cracking events, a micro-mechanical analysis of cross-ply laminates is herein proposed. Particularly, the cross-ply laminates belonging to the family [0°2/90°n/0°2] are considered via the generation of high-fidelity micro-mechanical models, which reproduce the internal fiber arrangements related to the reference results in Saito et al. 2012, Experimental evaluation of the damage growth restraining in 90° layer of thin-ply CFRP cross ply laminates, Adv. Comp Mat, 21:1,57–66. Differing from alternative approaches, current predictions are equipped with the combined used of two fracture-based modeling methods: (i) the variational phase field (PF) approach for triggering crack events into the matrix, and (ii) the bilinear cohesive zone model (CZM) for the simulation of fibre-matrix decohesion failures. Relying on this computational methodology, the focus of this work is to analyze the influence of the transverse ply thickness (n) on the onset and propagation of damage under tensile conditions, in conjunction with the transition from micro-cracking to meso-scale damage states. For this purpose, several models are generated into the FE package ABAQUS using user-defined capabilities, replicating configurations of specimens included in previous experimental investigations, i.e. through the consideration of different transverse ply thickness (n = 1, 2, 4). Present results show the potential of the proposed methodology to predict the transverse matrix cracking phenomena and the ability and reliability for capturing the delay of the crack with thinner plies, a phenomenon which is usually denominated as in-situ effect in the related literature.

Variational Phase Field Formulations of Polarization and Phase Transition in Ferroelectric Thin Films

The electric field plays an important role in ferroelectric phase transition. There have been numerous phase field formulations attempting to account for electrostatic interactions subject to different boundary conditions. In this paper, we develop new variational forms of the phase field electrostatic energy and the relaxation dynamics of the polarization vector that involves a hybrid representation in both real and Fourier variables. The new formulations avoid ambiguities appearing in earlier studies and lead to much more effective ways to perform variational studies and numerical simulations. Computations of phase transition and polarization switching in a single domain by applying the new formulations are provided as illustrative examples.

Role of interfacial transition zone in phase field modeling of fracture in layered heterogeneous structures

Mechanical behavior of layered materials and structures greatly depends on the mechanical behavior of interfaces. In the past decades, the failure in such layered media has been studied by many researchers due to their critical role in the mechanics and physics of solids. This study aims at investigating crack-interface interaction in two-dimensional (2-D) and three-dimensional (3-D) layered media by a phase field model. Our objectives are fourfold: (a) to better understand fracture behavior in layered heterogeneous systems under quasi-static load; (b) to introduce a new methodology for better describing interfaces by a regularized interfacial transition zone in the context of variational phase field approach, exploring its important role; (c) to show the accuracy, performance and applicability of the present model in modeling material failure at the interfaces in both 2-D and 3-D bodies; and (d) to quantitatively validate computed crack path with respect to experimental data. Phase field models with both perfectly and cohesive bonded interfaces are thus derived. A regularized interfacial transition zone is introduced to capture characteristics of material mismatch at the interfaces. Numerical examples for 2-D and 3-D layered systems with experimental validation provide fundamentals of fracture behavior in layered structures. The obtained results shed light on the behavior of crack paths, which are drastically affected by the elastic modulus mismatch between two layers and interface types, and reveal the important role of the proposed interfacial transition zone in phase field modeling of crack-interface interactions.