Phase-field Evolution Equation Research Articles

Multi-phase-field approach to fracture demonstrating the role of solid-solid interface energy on crack propagation

AbstractA multi-phase-field approach for crack propagation considering the contribution of the interface energy is presented. The interface energy is either the grain boundary energy or the energy between a pair of solid phases and is directly incorporated into to the Ginzburg–Landau equation for fracture. The finite difference method is utilized to solve the crack phase-field evolution equation and fast Fourier method is used to solve the mechanical equilibrium equation in three dimensions for a polycrystalline material. The importance of the interface (grain boundary) energy is analyzed numerically for various model problems. The results show how the interface energy variations change the crack trajectory between the intergranular and transgranular fracture.

Study on frost heaving force and the propagation of cracks in freezing rocks using the phase field method

In cold regions, the water in rocks expands volumetrically after freezing, leading to the propagation of fractures. In this study, the effective volume expansion coefficient is employed as a substitute for the volume expansion strain after the water in the fracture undergoes freezing and transforms into ice, and the phase field method is used to simulate the propagation of fractures in freezing rocks under various conditions to further explore the law of fracture propagation. The Abaqus software is used to solve the mechanical field control equation and the phase field evolution equation. First, the theoretical calculation of the frost-heaving pressure is compared with that obtained by experimental and numerical calculations. This comparison indicates that the numerical model presented in this study can effectively calculate the frost-heaving pressure during fracture propagation. Subsequently, the propagation of single and double fractures caused by frost heave is simulated based on fracture propagation experiments under frost heave conditions. The simulation results closely resemble the observed morphology of fracture propagation caused by frost heave obtained from the experiments. This confirms the suitability of the proposed model for simulating and predicting the propagation of fractures caused by frost heave. Finally, the influence of volume expansion strain growth path on fracture propagation due to frost heave is discussed. Although the final volume expansion strain value is the same, different growth paths of the volume expansion strain result in different fracture propagation morphologies.

Coupled thermo-hydro-mechanical-phase field modelling for hydraulic fracturing in thermo-poroelastic media

This study presents a coupled thermo-hydro-mechanical-phase field model, grounded in a thermodynamic framework, to elucidate the process of hydraulic fracture evolution in thermo-poroelastic media. The thermo-poroelastic characteristics of the rock exhibit variations contingent upon the phase field value. Consequently, the energy stored within the fluid and rock becomes a function of the phase field values, leading to the formulation of novel phase field evolution equations. For discretizing the thermal-hydrodynamic-mechanical-phase field model, the finite element method is employed. An interleaved scheme is utilized to concurrently solve for displacement, pressure, phase field, and temperature. The corresponding iterative structure is established using the Newton–Raphson method. Several hydraulic fracturing scenarios are investigated to showcase the potential and versatility of the presented model. These scenarios include: a) investigating the impact of different reservoir temperatures on hydraulic fracture propagation, b) analyzing the effects of intersection angle and reservoir temperature on fracture propagation in naturally fractured reservoirs, and c) examining the influence of reservoir temperature on hydraulic fracture propagation in grain-scale polymineral rock.

Mesoscale simulations of spherulite growth during isothermal crystallization of polymer melts via an enhanced 3D phase-field model

A finite difference-based 3D phase-field model is developed to investigate the spherulite growth at the mesoscopic scale during the isothermal crystallization of polyamide (PA) 12. The model introduces a phase-field variable to distinguish the crystalline and amorphous phases of polymers. The phase-field evolution equation is coupled with the heat conduction equation that considers the latent heat of crystallization. The evolution equations introduce both the dimensionless diffusivity and latent heat that are dependent on the crystallization temperature. A high-order finite difference-based numerical framework is applied to the phase-field model. Both the qualitative simulation results of the phase-field model such as the crystal morphologies and the quantitative results including the radial crystal growth rate, degree of crystallinity, and lamellar thickness are validated against experiments. The simulation for single-crystal growth shows that a high crystallization temperature results in a large crystal with a slow radial growth rate. The simulation for multi-crystal growth shows that the crystals impinge on each other and finally fill the whole domain during crystallization, which further demonstrates the capability of the model in simulating the spherulite growth during isothermal crystallization of polymer melts.

Simplified and complete phase-field fracture formulations for heterogeneous materials and their solution using a Fast Fourier Transform based numerical method

This paper focuses on the numerical implementation of phase-field models of fracture using the Fast Fourier Transform based numerical method. Recent developments in that field rely on the separate solution of a coupled problem where the mechanical equilibrium problem is solved first, and then the phase-field evolution equation. The latter involves a diffusion term which has been simplified in previous works relying on the Fast Fourier Transform based numerical method. This simplification has theoretically no effect for homogeneous materials, but might influence predictions significantly for heterogeneous materials where fracture properties vary between the different components.In this paper, the influence of this simplification is assessed and a complete formulation is proposed as well as a novel implementation of this formulation using the Fast Fourier Transform based numerical method. The assessment relies on simulations with a material containing two components, one of them being defined as unbreakable by using higher fracture properties. Using the simplified formulation, the presence of an artificial diffusion of damage between the two components is evidenced, and non-zero damage values are observed in the unbreakable component. Although the complete formulation leads to an increase of the number of iterations to solve the phase-field evolution equation, it suppresses completely the diffusion of damage towards the unbreakable component. The two formulations, in fact, lead to identical results when the fracture properties are homogeneous, but the results diverge both in terms of local fracture patterns and global stress–strain relations when the fracture properties contrast increases. This difference is also more pronounced when the regularization length introduced by the phase-field model increases.

An adaptive local algorithm for solving the phase-field evolution equation in the phase-field model for fracture

In the phase-field model for fracture, material damage can be characterized by a variable called the crack phase-field. Usually-two sub-problems controlled by the equilibrium equation and the evolution equation, respectively, are needed to be solved in the whole domain. In this paper, an adaptive local algorithm is proposed for the unified phase-field model to solve the evolution equation in the local domain based on the fact that the value of phase-field is non-zero only around the crack. In the initial unified phase-field model, the boundedness and irreversibility conditions are not easy to be satisfied. In this paper, a history field is proposed to convert the inequality into equality in the evolution equation, and the boundedness and irreversibility conditions can be treated easily as in the classical phase-field model. To further obtain the local algorithm, some failure criteria are introduced to identify the damaged area, and the elements in the local domain are updated adaptively in each iteration step. The local algorithm can be implemented easily since the boundary conditions of the local domain are the same as in the non-local algorithm. Numerical examples have shown that the proposed local algorithm can obtain the same results as the non-local algorithm.

A rate-dependent phase-field framework for the dynamic failure of quasi-brittle materials

In recent years, phase-field theory has become an efficient approach for predicting damage and fracture of engineering materials. However, the rate-dependence of quasi-brittle materials is not considered in most exiting models. To address this issue, we develop a new phase-field damage model for the dynamic failure of quasi-brittle solids based on the microforce balance law, within which a linear viscoelastic constitutive relation in effective stress space and a hyperbolic phase-field evolution equation are incorporated. Then several representative numerical examples are presented to demonstrate the ability of the proposed model in characterizing the dynamic failure process of quasi-brittle solids. Good agreements are achieved between numerical predictions and theoretical and experimental results. In particular, the increase of tensile strength and the transition of failure modes of concrete-like materials with the increase of loading rates can be well reproduced.

Computational homogenisation of phase-field fracture

In this manuscript, the computational homogenisation of phase-field fractures is addressed. To this end, a variationally consistent two-scale phase-field fracture framework is developed, which formulates the coupled momentum balance and phase-field evolution equations at the macro-scale as well as at the Representative Volume Element (RVE11The term ‘RVE’ is used interchangeably with sub-scale domain or microstructure, in this manuscript.) scale. The phase-field variable represent fractures at the RVE scale, however, at the macro-scale, it is treated as an auxiliary variable. The latter interpretation follows from the homogenisation of the phase-field through volume or a surface-average. For either homogenisation choices, the set of macro-scale and sub-scale equations, and the pertinent macro-homogeneity satisfying boundary conditions are established. As a special case, the concept of selective homogenisation is introduced, where the phase-field is chosen to live only in the RVE domain, thereby eliminating the macro-scale phase-field evolution equation. Numerical experiments demonstrate the local macro-scale material behaviour of the selective homogenisation based two-scale phase-field fracture model, while its non-selective counterpart yields a non-local macro-scale material behaviour.

A combined finite element–finite volume framework for phase-field fracture

Numerical simulations of brittle fracture using phase-field approaches often employ a discrete approximation framework that applies the same order of interpolation for the displacement and phase-field variables. In particular, the use of linear finite elements to discretize both stress equilibrium and phase-field equations is widespread in the literature. However, the use of P1 Lagrange shape functions to model the phase-field is not optimal, as the latter contains cusps for fully developed cracks. These should in turn occur at locations corresponding to Gauss points of the associated FE model for the mechanics. Such a feature is challenging to reproduce accurately with low order elements, and element sizes must consequently be made very small relative to the phase-field regularization parameter in order to achieve convergence of results with respect to the mesh. In this paper, we combine a standard linear FE discretization of stress equilibrium with a cell-centered finite volume approximation of the phase-field evolution equation based on the two-point flux approximation constructed over the same simplex mesh. Compared to a pure FE formulation utilizing linear elements, the proposed framework results in looser restrictions on mesh refinement with respect to the phase-field length scale. This ability to employ coarser meshes relative to the traditional implementation allows for significant reductions on computational cost, as demonstrated in several numerical examples.

A consistent phase field model for hydraulic fracture propagation in poroelastic media

We present a novel phase field method for modeling hydraulic fracture propagation in poroelastic media. In this approach, a new phase field evolution equation is derived to account for damage dependent poro-elastic parameters (Biot’s coefficient, Biot’s modulus and porosity). The fluid flow obeys Darcy’s seepage law in the entire domain including the damage zone, where the rock permeability is assumed to be anisotropic, following the maximum principal strain. The fully coupled problem is solved by a staggered scheme in which the mechanical equilibrium and fluid flow equations are linearized and solved using a Newton–Raphson(NR) method. Several numerical results are presented to investigate the effectiveness of the proposed formulation. First, stability and convergence of the method are verified on a set of benchmark problems considering different time steps and mesh sizes. Second, it is shown that if the poroelastic parameters are kept constant and do not change with the phase field parameter, i.e. reducing to standard phase field approaches in the literature, the model will tend to underestimate the fracture length and overestimate the pore pressure. Finally, we study the interaction of a propagating hydraulic fracture in porous media with inclined natural fractures, and simulate the hydraulic fracture propagation with different perforation phase angle.

Concepts of modeling surface energy anisotropy in phase-field approaches

To simulate the growth of geological veins, it is necessary to model the crystal shape anisotropy. Two different models, classical and natural models, which incorporate the surface energy anisotropy into the objective functional of Ginzburg–Landau type, are presented here. Phase-field evolution equations, considered in this work, are derived using the variational approach, and correspond to the conservative Allen–Cahn-type equation. For three characteristic anisotropy formulations, we show what kind of difficulties arise in the simulations for the presented models. Particularly, if the anisotropy becomes strong, the phase-field evolution equations become ill-posed. Thus, we present regularized phase-field models and discuss the corresponding simulation results. Furthermore, in the scope of the grain growth simulation, we extend the original two-phase models to multiphases.

Phase field modeling of fracture in anisotropic brittle solids

A phase field model of fracture that accounts for anisotropic material behavior at small and large deformations is outlined within this work. Most existing fracture phase field models assume crack evolution within isotropic solids, which is not a meaningful assumption for many natural as well as engineered materials that exhibit orientation-dependent behavior. The incorporation of anisotropy into fracture phase field models is for example necessary to properly describe the typical sawtooth crack patterns in strongly anisotropic materials. In the present contribution, anisotropy is incorporated in fracture phase field models in several ways: (i) Within a pure geometrical approach, the crack surface density function is adopted by a rigorous application of the theory of tensor invariants leading to the definition of structural tensors of second and fourth order. In this work we employ structural tensors to describe transverse isotropy, orthotropy and cubic anisotropy. Latter makes the incorporation of second gradients of the crack phase field necessary, which is treated within the finite element context by a nonconforming Morley triangle. Practically, such a geometric approach manifests itself in the definition of anisotropic effective fracture length scales. (ii) By use of structural tensors, energetic and stress-like failure criteria are modified to account for inherent anisotropies. These failure criteria influence the crack driving force, which enters the crack phase field evolution equation and allows to set up a modular structure. We demonstrate the performance of the proposed anisotropic fracture phase field model by means of representative numerical examples at small and large deformations.

Lattice Boltzmann simulations of 3D crystal growth: Numerical schemes for a phase-field model with anti-trapping current

A lattice-Boltzmann (LB) scheme, based on the Bhatnagar–Gross–Krook (BGK) collision rules is developed for a phase-field model of alloy solidification in order to simulate the growth of dendrites. The solidification of a binary alloy is considered, taking into account diffusive transport of heat and solute, as well as the anisotropy of the solid–liquid interfacial free energy. The anisotropic terms in the phase-field evolution equation, the phenomenological anti-trapping current (introduced in the solute evolution equation to avoid spurious solute trapping), and the variation of the solute diffusion coefficient between phases, make it necessary to modify the equilibrium distribution functions of the LB scheme with respect to the one used in the standard method for the solution of advection–diffusion equations. The effects of grid anisotropy are removed by using the lattices D3Q15 and D3Q19 instead of D3Q7. The method is validated by direct comparison of the simulation results with a numerical code that uses the finite-difference method. Simulations are also carried out for two different anisotropy functions in order to demonstrate the capability of the method to generate various crystal shapes.

Phase Field Approximation of Dynamic Brittle Fracture

AbstractThe numerical assessment of fracture has gained importance in fields like the safety analysis of technical structures or the hydraulic fracturing process. The modelling technique discussed in this work is the phase field method which introduces an additional scalar field. The smooth phase field distinguishes broken from undamaged material and thus describes cracks in a continuum. The model consists of two coupled partial differential equations ‐ the equation of motion including the constitutive behaviour of the material and a phase field evolution equation. The crack growth follows implicitly from the solution of this system of PDEs. The numerical solution with finite elements can be accelerated with an algorithm that performs computationally extensive tasks on a graphic processing unit (GPU). A numerical example illustrates the capability of the model to reproduce realistic features of dynamic brittle fracture. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase-field modeling of the microstructure evolution and heterogeneous nucleation in solidifying ternary Al–Cu–Ni alloys

We have investigated the microstructure evolution during the isothermal and non-isothermal solidification of ternary Al–Cu–Ni alloys by means of a general multi-phase-field model for an arbitrary number of phases. The stability requirements for the model functions on every dual interface guarantee the absence of “ghost” phases. The aim was to generate a realistic microstructure by coupling the thermodynamic parameters of the phases and the thermodynamically consistent phase-field evolution equations. It is shown that the specially constructed thermal noise terms disturb the stability on the dual interfaces and can produce heterogeneous nucleation of product phases at energetically favorable points. Similar behavior can be observed in triple junctions where the heterogeneous nucleation of a fourth phase is more favorable. Finally, the model predicts the growth of a combined eutectic-like and peritectic-like structure that is comparable to the observed experimental microstructure in various alloys.

MPI + OpenCL implementation of a phase-field method incorporating CALPHAD description of Gibbs energies on heterogeneous computing platforms

Phase-field method uses a non-conserved order parameter to define the phase state of a system and is a versatile method for moving boundary problems. It is a method of choice for simulating microstructure evolution in the domain of materials engineering. Solution of phase-field evolution equations avoids explicit tracking of interfaces and is often implemented on a structured grid to capture microstructure evolution in a simple and elegant manner. Restrictions on the grid size to accurately capture the interface curvature effects lead to large number of grid points in the computational domain and render the simulation computationally intensive for realistic simulations in 3D. However, the availability of powerful heterogeneous computing platforms and super clusters provides the advantage to perform large scale phase-field simulations efficiently. This paper discusses a portable implementation to extend simulations across multiple CPUs using MPI to include use of GPUs using OpenCL. The solution scheme adapts an isotropic stencil that avoids grid-induced anisotropy. Use of separate OpenCL kernels for problem specific portions of the code ensure that the approach can be extended to different problems. Performance analysis of parallel strategies used in the study illustrate the massively parallel computing possibility for phase-field simulations across heterogeneous platforms.

On the incorporation of cubic and hexagonal interfacial energy anisotropy in phase field models using higher order tensor terms

In this paper, we show how to incorporate cubic and hexagonal anisotropies in interfacial energies in phase field models; this incorporation is achieved by including up to sixth rank tensor terms in the free energy expansion, assuming that the free energy is only a function of coarse-grained composition, its gradient, curvature and aberration. We derive the number of non-zero and independent components of these tensors. Further, by demanding that the resultant interfacial energy is positive definite for inclusion of each of the tensor terms individually, we identify the constraints imposed on the independent components of these tensors. The existing results in the invariant group theory literature can be used to simplify the process of construction of some (but not all) of the higher order tensors. Finally, we derive the relevant phase field evolution equations and describe some preliminary results from our 1D simulations.

Towards a thermodynamically consistent picture of the phase-field model of vesicles: Local membrane incompressibility

A phase-field model for vesicles including hydrodynamics was presented in two and three dimensions [T. Biben and C. Misbah, Phys. Rev. E 67, 031908 (2003); T. Biben, K. Kassner, and C. Misbah, Phys. Rev. E 72, 041921 (2005)]. A particularly important feature for vesicles is that their membrane is locally incompressible. In these works a tension field defined everywhere in the bulk was introduced in order to fulfill local membrane inextensibility. Here we reconsider the original model by treating the phase field as a thermodynamic variable and develop a picture which is consistent with the second law of thermodynamics. This enables us to write the phase-field evolution equations in terms of a thermodynamical potential. This potential acquires, at global equilibrium, a Lyapunov functional character. The goal of this paper is twofold: (i) The first and primary goal is purely conceptual, in that we can write down a first and second principle for membranes, from which the evolution equations follow, thanks to the evaluation of the entropy production and the use of concepts of irreversible thermodynamics. (ii) Due to the monotonous character of the evolution of the functional (at global equilibrium), we expect this formulation to be more appropriate for numerical studies. The formalism developed to account for the local incompressibility of the membrane is believed to offer a systematic framework in order to include naturally other physical ingredients, as briefly discussed here and demonstrated in future works.

Phase Field Modeling of Surface Instabilities Induced by Stresses

ABSTRACTIn this work, we developed a phase field model describing surface evolution dynamics of a strained solid in contact with gas phase. Elastic solutions are solved with an iteration method for the system in which a solid film, strained by the mismatch strain between the film and substrate, has anisotropic elastic constants and the vapor phase has zero elastic constants. Elastic solutions are incorporated into the phase field evolution equation. The model predicts stable morphology at a given mismatch strain and perturbation wavelength.

Phase-field analyses for Y123 crystal growth

A phase-field evolution equation (a kind of time dependent Ginzburg–Landau equation), in which order parameter ( φ) transition in width restricted within a unit grid, was solved using an implicit finite difference method. The direct implicit scheme would have an ability to correspond to a long-range order parameter correlation, which appeared around a phase transition. When the transition width is restricted within a unit grid, a skew directional finite difference is necessary to obtain an isotropic growth rate. Using these schemes, an isotropic growth shape was obtained for a single composition system with a latent heat free condition. In the case of a large latent heat condition, which is the case for Y123 crystal growth, we could obtain a faceted growth shape without using any anisotropic parameters.