Phase-field Variable Research Articles

A phase-field method for elastic mechanics with large deformation

Boundary tracking is a challenging numerical problem for elastic mechanics problems with large deformation. In particular, when the boundary conditions rely on the boundary shape, iteration is required between tracking boundaries and solving the elasticity equations. Stability of this iteration brings additional difficulty for designing efficient numerical methods. In this work, we introduce phase-field variables to represent different elastic domains and track the movement of boundaries. Furthermore, boundary conditions relying on the boundary shape can also be easily applied with the phase-field variables. Second-order compact finite difference schemes are used to solve the coupled system on a square-mesh staggered grid. Our numerical results show that the phase-field method can be conveniently used to solve complex elastic mechanics problems accurately.

Applications of a phase-field fracture model to materials with inclusions

Fracture in materials with inclusions is considered in the computational model of this contribution which allows prediction of crack propagation. The inclusions cause that cracks may appear inside the materials or along matrix-inclusion interfaces. The presented model can treat them both using two internal variables in a fashion of damage mechanics so that fracture is a consequence of the damage. The first variable is defined at the interface represented by a thin degradable adhesive layer so that an appropriate stress-strain relationship can be obtained as in cohesive zone models. The second variable is defined in the structural domains as a phase-field fracture variable which causes elastic properties degradation in a narrow material strip that forms a smeared crack. Both these damaging schemes are expressed in a unique quasi-static energy evolution process. The numerical solution approach is thus rendered from a variational form with a staggered time stepping procedure related to a separation of deformation variables from the damage ones and using non-linear programming algorithms implemented together within an own MATLAB finite element code. The numerical simulations with the model include simplified structural and material elements.

A mass-conserving and volume-preserving lattice Boltzmann method with dynamic grid refinement for immiscible ternary flows

In this paper, a lattice Boltzmann model with dynamic grid refinement is proposed for immiscible ternary flows, which is capable of conserving the total mass and preserving the volume of each phase. The application of interpolation schemes in adaptive mesh refinement (AMR) techniques results in the violation of the total mass of the fluids system within the lattice Boltzmann method (LBM) framework. In the present model, a source term with two free parameters is introduced into the interface capturing equation, which can be determined by the mass conservation and the volume preservation properties. A piecewise constant function has been deliberately incorporated into the source term in order to avoid the appearance of an unphysical fluid at the interface of other two fluids. Based on a block-structured AMR method, the governing equations for phase-field variables and flow hydrodynamic properties are solved by the finite-difference multiple-relaxation-time (MRT) LBM. Simulations of several typical problems are performed in order to evaluate the accuracy and applicability of the proposed model. The numerical results demonstrate that the present model can conserve both mass and volume at the same time as well as reduce numerical dispersion in the bulk region.

Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. Specifically in the optimal control minimization problem, a tracking-type cost functional is minimized to steer the crack via the phase-field variable into a desired pattern. To achieve such optimal solutions, Neumann type boundary conditions need to be determined. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint, namely the displacement and phase-field unknowns, but keeping the control variable as the only unknown. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Both our formulation and algorithmic developments are substantiated and illustrated with six numerical experiments.

Modelling and simulation of fracture in anisotropic brittle materials by the phase-field method with novel strain decompositions

When brittle and quasi-brittle materials, are concerned, it is essential to account for the fundamental fact that they behave differently according as they undergo tensile or compressive loadings. Due to this fact, in modelling and simulating the fracture of brittle and quasi-brittle materials by the phase-field method (PFM), the strain tensor is usually decomposed into a tensile part and a compressive part, and it is argued that only the strain energy related to the tensile part controls the nucleation and propagation of cracks. The present work improves the phase-field method by incorporating into it a novel strain decomposition strategy giving rise to strain decompositions being orthogonal in the sense of the natural inner product with the elastic stiffness tensor acting as a metric. The improved phase-field method involving only a scalar phase-field variable is applied to model and simulate the fracture of brittle anisotropic materials. The results thus obtained are compared with the relevant analytical and numerical ones reported in the literature. It turns out from these comparisons that the improved phase-field method is accurate and efficient.

Phase-field modeling for dynamic cracking behavior in bedded shale

The bedding plane with the special structural characteristics may have a significant influence on the cracking behavior induced by stress wave in bedded shale. This study proposes a novel phase-field model to predict the dynamic crack propagation path in bedded shale. The bedding plane is assumed to be the thin elastic layer with the certain thickness and independent mechanical parameters. Because its thickness is tiny compared with rock matrix, the lower-dimensional interfacial element is employed to discretize it together with rock matrix within finite element framework. The fractures within bedding plane and rock matrix share a phase-field variable. However, the governing equations of the phase-field evolutions in bedding plane and rock matrix have different driving forces, and such operation greatly increases the robustness of the model. Based on this model, the dynamic cracking behavior of bedded shale is investigated numerically. The results indicate that the mechanical properties of bedding planes are the key factors affecting the overall cracking behavior of bedded shale. Our research provides a valuable insight in cracking behavior of bedded shale in dynamic fracturing for shale gas development.

Study on the Characteristics of Water Jet Breaking Coal Rock in a Drilling Hole

Water jet technology is an effective measure by which to improve the efficiency of deep coalbed methane mining. Nevertheless, the effect of a water jet impinging on coal rock remains unclear. In this study, numerical simulation is used to analyze a water jet impingement drill test block, after which experimental verification is carried out. Next, on the basis of the experimental verification that the simulation method is feasible, the influence factors of the water jet impingement in the drill hole are analyzed. It is concluded that the phase-field variables based on the fracture change method can effectively characterize the damage and destruction of coal rock. The water jet impact in the borehole has a central damage failure zone and two-side damage failure zone, and the damage failure ratio n is used to characterize the degree of damage to the coal rock. When the jet target distance is 70 mm, the damage ratio n is closest to 1, and the effect of water jet impact on coal-rock is the best. When the wall roughness is less than 10 mm, the blocking effect on the jet is dominant, resulting in a negative correlation between the damage size and the roughness. When the wall roughness exceeds 10 mm, the development of cracks and the connection effect exceed the blocking effect, resulting in larger damage. Therefore, it can be seen that the effect of a water jet impacting on coal rock is positively related to jet pressure. When the jet target distance is 70 mm, the damage range of 30 MPa jet pressure to the center and both sides of the coal rock reaches about 1.1 m, thus signifying a good coal rock breaking effect. The wall roughness has a significant effect on the coal rock breaking effect of the water jet.

A three-field phase-field model for mixed-mode fracture in rock based on experimental determination of the mode II fracture toughness

In this contribution, a novel framework for simulating mixed-mode failure in rock is presented. Based on a hybrid phase-field model for mixed-mode fracture, separate phase-field variables are introduced for tensile (mode I) and shear (mode II) fracture. The resulting three-field problem features separate length scale parameters for mode I and mode II cracks. In contrast to the classic two-field mixed-mode approaches, it can thus account for different tensile and shear strength of rock. The two phase-field equations are implicitly coupled through the degradation of the material in the elastic equation, and the three fields are solved using a staggered iteration scheme. For its validation, the three-field model is calibrated for two types of rock, Solnhofen Limestone and Pfraundorfer Dolostone. To this end, double-edge notched Brazilian disk (DNBD) tests are performed to determine the mode II fracture toughness. The numerical results demonstrate that the proposed phase-field model is able to reproduce the different crack patterns observed in the DNBD tests. A final example of a uniaxial compression test on a rare drill core demonstrates that the proposed model is able to capture complex, 3D mixed-mode crack patterns when calibrated with the correct mode I and mode II fracture toughness.

Explicit double-phase-field formulation and implementation for bending behavior of UHPC-NC composite beams

The Phase-field model emerges as a promising method since it can automatically capture the initiation, propagation, coalescence, and branching of cracks without any tracking algorithms. Despite its success in simulating brittle and quasi-brittle cracks, tensile (mode I) and shear (mode II) fractures cannot be distinguished effectively. Moreover, the phase-field model is rarely applied to specimens at the structural level (i.e., reinforced concrete beams), due to the demanding computational cost. To address these shortfalls, two sets of phase-field variables are formulated to describe the tensile and shear fractures of concrete, respectively, through crack-direction-based decomposition of strain energy density into tensile, shear and compressive parts. Computational burdens are greatly alleviated by implementing the model with an explicit finite element solver (Abaqus/Explicit), allowing for massive parallel computing. The sensitivity analysis is performed to investigate the effects of critical parameters with a 1D numerical tensile bar. The double-phase-field model is applied to study the bending behavior of composite beams consisting of normal-strength concrete (NC) overlayer and ultra-high performance concrete (UHPC) substrate. To this end, five UHPC-NC beams, with various thickness ratios of UHPC and NC layers, were prepared and loaded to failure. District bending strengths, cracking patterns, deformation characteristics and failure modes of the four beams were revealed and discussed. It shows that the proposed model is necessary to accurately capture the bending behavior of UHPC-NC composite beams. This model is also applied to discuss the effects of critical parameters on the bending behaviors of UHPC-NC composite beams. To benefit potential users, detailed explanations regarding the data structures of input files and user subroutines (VUEL, VUMAT, VUSDFLD in Abaqus/Explicit) are given and released on GitHub repository.

A hybrid multi-phase field model to describe cohesive failure in orthotropic materials, assessed by modeling failure mechanisms in wood

Fracture mechanics is crucial for many fields of engineering applications, as precisely predicting failure of structures and parts is required for efficient designs. The simulation of failure processes is, both from a mechanical and a numerical point of view, challenging, especially for inhomogeneous materials, where the microstructure influences crack initiation and propagation and might lead to very complex crack patterns. The phase field method for fracture is a promising approach to encounter such materials, since it is able to describe complex fracture phenomena like crack kinking, branching and coalescence. Moreover, it is a largely mesh independent approach, given that the mesh is homogeneous in the area of the crack. However, the original formulation of the phase field method is limited to isotropic materials and does not account for preferable fracture planes defined through the material’s microstructure. In this work, the method is expanded to take orthotropic constitutive behavior and preferable directions of crack propagation into account. We show that by using a stress-based split and multiple phase field variables with preferable fracture planes, in combination with a hybrid phase field approach, a general framework can be found for simulating anisotropic, inhomogeneous materials. The stress-based split is based on fictitious crack faces and is, herein, expanded to support anisotropic materials. Furthermore, a novel hybrid approach is used, where the degradation of the sound material is performed based on a smooth traction free crack boundary condition, which proves to be the main driving factor for recovering commonly observed crack patterns. This is shown by means of a detailed analysis of two examples: a wooden single edge notched plate and a wood board with a single knot and complex fiber directions. In both cases, the proposed novel hybrid phase field approach is able to realistically reproduce complex failure modes.

Chemo-mechanical modeling of inter- and intra-granular fracture in heterogeneous cathode with polycrystalline particles for lithium-ion battery

Evolution of complex fracture patterns during electrochemical cycling for the polycrystalline particulate cathode of lithium-ion batteries is found to be a primary reason towards capacity fading. Such fracture of a particles involves inter- and intra-granular crack propagation and often with its debonding from the binder. Till date, a comprehensive understanding of these fracture mechanisms under chemo-mechanical environment towards overall electrochemical response is lacking. In that context, a thermodynamically consistent multi-physics modeling framework is developed to account for strongly coupled chemo-mechanical environments and the evolution of arbitrary fracture within polycrystalline cathode particles embedded in the matrix. Specifically, a regular phase-field fracture variable is introduced for initiation and propagation of the inter- and intra-granular crack within the particle and debonding of the particle-matrix interface. Adopting two additional phase-field descriptors, grain boundaries and particle-matrix interfaces are expressed as diffused regimes. Thus, fracture energies of the grains, grain boundaries, and particle-matrix interface are suitably incorporated within the mathematical framework. Employing a finite element-based numerical scheme, the coupled governing equations are solved for three state variables: concentration, displacement, and phase-field damage. Various parametric studies are performed to investigate the effect of grain sizes, charge rates, and elastic modulus of the matrix on the interplay between the chemo-mechanical behavior and evolution of fracture, and ultimately their collective influences on the electrochemical response. Finally, the present findings offer predictive insights towards engagement of key fracture mechanisms for mechanical degradation of the polycrystalline cathode at electrode level.

Multi phase-field modeling of anisotropic crack propagation in 3D fiber-reinforced composites based on an adaptive isogeometric meshfree collocation method

An anisotropic multi phase-field model is developed in this paper to investigate the interaction between the crack propagation and interfacial damage in a fiber-reinforced composite. The proposed modeling approach can capture the fracture patterns of composite microstructures through the introduction of the crack phase field and the interface phase field. By considering the free energy related to the interface and crack phase fields, the interfacial debonding, matrix cracking and the interaction of two fracture patterns can thus be simulated. The anisotropic phase-field approach is further adopted to describe the interface interaction associated with a crack. The phase-field equations are solved using the isogeometric-meshfree collocation approach to achieve high computational efficiency. An adaptive h-refinement scheme is incorporated into the phase-field formulations using phase-field variables and their gradients as the error indicators. The proposed method is shown to be effective and robust in both case studies of 2D and 3D fiber-reinforced composite microstructures. Moreover, fracture behaviors including the crack initiation, propagation, coalescence, interfacial debonding, and matrix cracking in composite microstructures are found to be precisely modeled by the proposed approach.

An adaptive isogeometric phase-field method for brittle fracture in rock-like materials

In recent years, the phase-field method (PFM) has gained considerable attention to model fracture process. This is because, in a single framework, the method allows to model crack nucleation, growth, coalescence and branching. In addition, existing FE codes can be used with minimal modifications. However, one of the major limitations of the PFM is that, it requires extremely fine mesh to accurately capture the fracture process. An adaptive phase-field method within the framework of isogeometric analysis is presented in this work to model tensile and compressive-shear crack propagation in rock-like materials. Locally refined NURBS are adopted for describing both the geometry and the field variables. The local refinement is performed by structured mesh refinement strategy based on an error indicator that depends on the user-defined threshold value for the phase-field variable. Two phase-field models are adopted for simulating tensile and compressive-shear fractures in rock-like materials. For numerical implementation, a hybrid formulation combined with a staggered solution scheme is adopted. Several two-dimensional benchmark problems are investigated to demonstrated the robustness of the proposed framework. From the numerical study, it can be found that the developed adaptive framework achieves results comparable to a uniformly refined mesh with higher degrees of freedom.

Error analysis of a decoupled, linear and stable finite element method for Cahn–Hilliard–Navier–Stokes equations

In this paper, we carry out the error analysis for a totally decoupled, linear and unconditionally energy stable finite element method to solve the Cahn–Hilliard–Navier–Stokes equations. The fully finite element scheme is based on a stabilization for Cahn–Hilliard equation and projection method for Navier–Stokes equation, as well as the first order Euler method for time discretization. A priori error analysis for phase field, velocity field and pressure variable are derived for the fully discrete scheme.

Fracture analysis of single edge notched specimen using phase field approach

For the damage tolerant design of cracked structural components, two important parameters namely, residual strength and remaining life are very important. To evaluate these, a reliable fracture parameter such as stress intensity factor (SIF) is to be determined. Phase field (PF) based finite element formulation for a brittle fracture to determine crack propagation trajectory is presented in this paper. PF methodologies have been formulated by coupling partial differential equations (PDEs) after minimizing the total potential energy, which includes elastic strain energy and fracture energy, with respect to the displacement field and PF variable. Solutions of these PDEs for PF variable resulted in identifying crack path geometry, while PDEs displacement field obtained near the crack field is used to determine SIF. In the present study, extending PF formulation to fracture analysis of a structural component is first of its kind. To simulate PF phenomena, a four noded 2-D bilinear quadrilateral element with three degrees of freedom (DOF) at each node (two translations and one for phase field) has been developed to analyze single edge notched (SEN) specimen under tensile and shear loading. By combining the displacement results of PF method with displacement extrapolation technique, SIF has been determined for various crack lengths representing crack propagation in SEN specimens. It is then compared with the analytical SIF solution available in Handbook, and also with the contour integral method SIF results. It is found that the SIF obtained by PF method is in good agreement (±10%) with the other solutions.

An isogeometric phase–field based shape and topology optimization for flexoelectric structures

For considering topological changes in a CAD-based framework, an isogeometric phase-field based shape and topology optimization is developed. In this case, the diffuse interface, between a material and void phases, describes the boundaries of the geometry. The descent direction to minimize the objective function is found by using a bulk energy function, which is written in terms of the sensitivities of the objective function with respect to the design variables. The design variables are the local values of the phase-field variable. The evolution of the phase-field is determined by solving the time-dependent Allen–Cahn equation. The phase-field topology optimization approach is used to optimize flexoelectric structures, a problem for which we exploit the advantages of using NURBS to discretize the geometry and the solution fields. The C1 continuity required for the solution of the flexoelectric PDE is easily obtained using NURBS as basis functions. The adjoint method is used to compute the sensitivity analysis. The results of the optimal flexoelectric geometries are compared to the optimal geometries in case piezoelectric materials are used. Different geometries, boundary conditions and material configurations are considered to demonstrate the robustness of the method. Compared to pure piezoelectricity, the electric output of piezo-flexoelectric microbeams under bending is larger. In contrast, for a structure under compression, the minimized value of the objective function is larger for piezoelectricity. In general, the enhancement of the energy conversion factor is larger when flexoelectricity is considered together with piezoelectricity. The optimal results show the relevance of considering flexoelectricity at microscales (3∼5 μm).

Unconditionally stable numerical methods for Cahn-Hilliard-Navier-Stokes-Darcy system with different densities and viscosities

In this article we consider the numerical modeling and simulation via the phase field approach for coupled two-phase free flow and two-phase porous media flow of different densities and viscosities. The model consists of the Cahn-Hilliard-Navier-Stokes equations in the free flow region and the Cahn-Hilliard-Darcy equations in porous media that are coupled by several domain interface conditions. It is showed that the coupled model satisfies an energy law. Then we first propose a coupled unconditionally stable finite element method for solving this model and analyze the energy stability for this method. Furthermore, based on the ideas of pressure stabilization and artificial compressibility, we propose an unconditionally stable time stepping method that decouples the computation of the phase field variable, the velocity and pressure of free flow, the velocity and pressure of porous media, hence significantly reduces the computational cost. The energy stability of this decoupled scheme with the finite element spatial discretization is rigorously established. We verify numerically that our schemes are convergent and energy-law preserving. Numerical experiments are also performed to illustrate the features of two-phase flows in the coupled free flow and porous media setting.

Air bubble entrapment during drop impact on solid and liquid surfaces

The phase-field modeling (PFM) of water drop impact onto a dielectric hydrophobic parafilm surface is performed to explore air entrapment and its influence on deposition and rebound phenomena. Local and global characteristics of the drop impact are taken into account by using the combined Cahn-Hilliard and Navier-Stokes equations. The modeling results of water drop impact are directly compared with our experimental measurements in terms of maximum spreading distance, and air bubble size. The simulation results reveal that air can be trapped under the liquid drop during the initial impact as well as during the retraction phase at the center of the drop due to the closure of the liquid layer above a cavity. It is found that the drop diameter and the impact velocity play significant roles in the air entrapment phenomena. The probability of air bubble formation is higher at lower impact velocity and for larger drop size. The model is also capable of simulating the case of drop impact onto a water surface, and the results are validated using prior literature data. In addition, the influence of the phase-field variables and the mesh adaptation scheme on the PFM is studied and discussed. Thus, our findings provide new qualitative and quantitative insights into the influence of air entrapment on drop deposition onto hydrophobic and liquid surfaces.

Crack propagation in quasi-brittle materials by fourth-order phase-field cohesive zone model

A phase-field approach becomes a more popular candidate in modeling crack propagation. It uses a scalar auxiliary variable, namely a phase-field variable, to model a discontinuity zone in a continuity domain. Furthermore, the fourth-order phase-field approach produces a better convergence rate and more accurate solutions than the second-order one. However, it is available for modeling crack propagation in brittle material. This study addresses the fourth-order phase-field model combining the non-standard phase-field form with a cohesive zone model (CZM) to predict crack propagation in quasi-brittle material. A Cornelisson's softening law is used to capture the high precision of crack propagation prediction. The concrete material is considered as a quasi-brittle one. For computation efficiency using NURBS-based finite elements, Virtual Uncommon-Knot-Inserted Master-Slave (VUKIMS) technique is employed to derive a local refinement mesh. Numerical results are verified by the published ones from literature. It was found that the peak load and crack path are independent of the element size and insensitive to the length-scale number using the fourth-order phase-field CZM. Our proposed model shows the most significant advantage compared to the standard phase-field approach in terms of computational cost and solution accuracy.

Port Hamiltonian formulation of the solidification process for a pure substance: A phase field approach*

In this paper we suggest a Port Hamiltonian model of the solidification process of water, using the phase field approach. Firstly, the Port Hamiltonian formulation of the dynamics of the phase field variable, governed by the Allen-Cahn equation, is recalled. It is based on adding to the phase field variable, its gradient, and extending the system with its dynamics. Secondly, the model is completed by the energy balance equation for the heat conduction and the complete Port Hamiltonian model is derived. Thirdly an Algebro-differential Port Hamiltonian representation is suggested, where the Port Hamiltonian system is defined on a Lagrangian submanifold, allowing to use directly the variables defining the thermodynamical data.