Phase-field Evolution Research Articles

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.

Phase-field modeling of fracture and crack growth in friction stir processed pure copper

The present work aims to predict the fracture mechanism of a friction stir processed (FSPed) copper. In particular, a phase-field model integrated with finite element model is developed for ductile fracture modeling. The phase-field evolution governed by the nonlinear coupled system comprising the linear momentum equation and the diffusion-type equation is solved concurrently through a Newton–Raphson approach. The proposed fracture model is established by correlating the phase-field degradation function with a scalar measure of the plastic strain, and assuming that the fracture takes place once the accumulated plastic strain reaches a critical value. The numerical simulation results are validated with experimental investigations. The results show that the proposed model is capable of capturing the experimentally observed sequence of elastoplastic base material behavior, FSP deformation, and fracture phenomena in specimens.

Phase field modeling of hydraulic fracturing in porous media formation with natural fracture

In this study, a new coupled fluid flow and fracture phase field evolution model is proposed to simulate hydraulic fracture propagation in porous media with natural fracture. In the proposed model, fracture width is calculated by maximum principal strain criterion, and only when the strain exceeds the critical strain there will be cracks formed, the element permeability is obtained by weighted calculation of fracture permeability and matrix permeability. The coupled fluid flow and stress equilibrium nonlinear equations is solved by the Newton–Raphson (NR) method, and the phase field is solved by Picard iteration method. The model convergence and stability were validated by compared the results of different mesh size and time step cases, respectively. The effectiveness of the model is validated by comparing the fracture width obtained from our numerical model and analytical model. Based on our numerical model, we investigate the influence factors of hydraulic fracture propagation in porous media with natural fracture. The results indicate that the smaller the approaching angle, natural fracture strength and in-situ stress difference, the easier for natural fracture initiating and propagating. Nevertheless, only one wing of the natural fracture can be activated.

Phase field modeling of fracture in Quasi-Brittle materials using natural neighbor Galerkin method

Recently developed phase field models of fracture require a diffusive crack representation based on an introduction of a crack phase field. We outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles. In this work, the numerical implementation of the phase field model for brittle fracture using natural neighbor Galerkin method is presented. Phase field method diffuses the sharp discontinuity by introducing a regularized crack functional. The notion of regularized crack functional provides a basis for defining suitable convex dissipation functions which govern the evolution of the crack phase field. The use of natural neighbors to dynamically decide the compact support at a nodal point makes it truly mesh free and nonlocal. The method uses smooth non-polynomial type Sibson interpolants, which are C0 at a given node and C∞ everywhere else. This allows to capture the interface region in a phase field model more accurately with only few nodes. For a given accuracy, there is an improvement in the computational time with the use of natural neighbor interpolants. Several benchmark problems were solved to investigate the efficiency of the numerical implementation of phase field model using natural neighbor Galerkin method.

On the hydraulic fracturing in naturally-layered porous media using the phase field method

In the hydraulic fracturing of natural rocks, understanding and predicting crack penetrations into the neighboring layers is crucial and relevant in terms of cost-efficiency in engineering and environmental protection. This study constitutes a phase field framework to examine hydraulic fracture propagation in naturally-layered porous media. Biot's poroelasticity theory is used to couple the displacement and flow field, while a phase field method helps characterize fracture growth behavior. Additional fracture criteria are not required and fracture propagation is governed by the equation of phase field evolution. Thus, penetration criteria are not required when hydraulic fractures reach the material interfaces. The phase field method is implemented within a staggered scheme that sequentially solves the displacement, phase field, and fluid pressure. We consider the soft-to-stiff and the stiff-to-soft configurations, where the layer interface exhibits different inclination angles θ. Penetration, singly-deflected, and doubly-deflected fracture scenarios can be predicted by our simulations. In the soft-to-stiff configuration, θ=0° exhibits penetration or symmetrical doubly-deflected scenarios, and θ=15° exhibits singly-deflected or asymmetric doubly-deflected scenarios. Only the singly-deflected scenario is obtained for θ=30°. In the stiff-to-soft configuration, only the penetration scenario is obtained with widening fractures when hydraulic fractures penetrate into the soft layer.

Modeling the dynamic and quasi-static compression-shear failure of brittle materials by explicit phase field method

The phase field method is a very effective method to simulate arbitrary crack propagation, branching, convergence and complex crack networks. However, most of the current phase-field models mainly focus on tensile fracture problems, which is not suitable for rock-like materials subjected to compression and shear loads. In this paper, we derive the driving force of phase field evolution based on Mohr–Coulomb criterion for rock and other materials with shear frictional characteristics and develop a three-dimensional explicit parallel phase field model. In spatial integration, the standard finite element method is used to discretize the displacement field and the phase field. For the time update, the explicit central difference scheme and the forward difference scheme are used to discretize the displacement field and the phase field respectively. These time integration methods are implemented in parallel, which can tackle the problem of the low computational efficiency of the phase field method to a certain extent. Then, three typical benchmark examples of dynamic crack propagation and branching are given to verify the correctness and efficiency of the explicit phase field model. At last, the failure processes of rock-like materials under quasi-static compression load are studied. The simulation results can well capture the compression-shear failure mode of rock-like materials.

A non-isothermal phase field study of the shape memory effect and pseudoelasticity of polycrystalline shape memory alloys

In this paper, a non-isothermal phase field (PF) model is proposed to investigate the shape memory effect (SME) and pseudoelasticity (PE) of polycrystalline shape memory alloys (SMAs) with consideration of the latent heat effect. The latent heat release and absorption accompanying the phase transformation processes are explicitly considered by coupling the phase field evolution with latent heat conduction. A modified Gibbs free energy function is proposed to accommodate the continuously varying temperature. It is shown that the SME and PE of polycrystalline SMAs under different ambient temperatures can be well characterized with the proposed PF model within a unified framework. The PF simulation results are in accordance with the previously reported experimental results. The influences of various factors such as the ambient temperature, grain size, crystal orientation, and latent heat effect on the phase transformation process and mechanical responses as well as the temperature evolution of SMAs are systematically discussed by conducting PF simulations. Some important implications for the devise of elastocaloric cooling devices are provided.

A diffuse interface lattice Boltzmann model for thermocapillary flows with large density ratio and thermophysical parameters contrasts

A diffuse interface lattice Boltzmann model for thermocapillary flows with large density ratio and thermophysical parameters contrasts is developed in this paper. In this model, three distribution functions are used to describe the evolution of phase field, velocity field and temperature field. The conservative Allen-Cahn-based lattce Boltzmann equation which has good numerical stability in simulating multiphase flows at high density ratio is employed to capture the phase interface. The velocity-based lattice Boltzmann equation is utilized to capture the hydrodynamics with thermocapillary effect. In particular, a lattice Boltzmann scheme is proposed to solve the temperature field equation based on the diffuse interface concept, in which the source term is computed locally. Unlike previous lattice Boltzmann model for thermocapillary flows, the thermophysical parameters (heat capacitance and thermal conductivity) of two fluids are allowed to be different. The present model is validated by simulating several numerical examples. Numerical results indicate the reliability of proposed diffuse interface lattice Boltzmann model in simulating thermocapillary flows with large density ratio (up to 1000) and thermophysical parameters contrasts. Moreover, we also derive an unsteady solution for thermocapillary-driven flow in a heated microchannel with two superimposed planar fluids, which can be used to assess the numerical accuracy of numerical algorithm for thermocapillary flows.

Phase field modeling of fracture in nonlinearly elastic solids via energy decomposition

Phase-field models for fracture problems have attracted considerable attention in recent years, which are capable of tracking the discontinuities numerically, and also produce complex crack patterns in many applications. In this paper, a phase-field model for a general nonlinearly elastic material is proposed using a novel additive decomposition of strain energy. This decomposition has two parts: one is principal stretch related and the other solely composed of volumetric deformation, which accounts for different behaviors of fracture in tension and compression. We construct the Lagrangian by integrating the split energies and the separation energy from phase-field approximation for discrete cracks. A coupled system of equations is also derived that governs the deformation of the body and the evolution of phase field. The capability and performance of the proposed model are demonstrated in several representative examples. Our results show that the predicted fracture surfaces are in good agreement with experimental observations. Compared with the previous models in which the energy is simply split into the isochoric and volumetric parts, the present model is numerically more robust and effective in simulating sharp cracks. The present model can also aid researchers to control the degree of tension–compression asymmetry in the nonlinear regime of deformation, which can be naturally extended to simulate the fracture of the rubber-like materials with tension–compression asymmetry.

Phase field modeling of quasi-static and dynamic crack propagation: COMSOL implementation and case studies

The phase-field model (PFM) represents the crack geometry in a diffusive way without introducing sharp discontinuities. This feature enables PFM to effectively model crack propagation compared with numerical methods based on discrete crack model, especially for complex crack patterns. Due to the involvement of “phased field”, phase-field method can be essentially treated a multifield problem even for pure mechanical problem. Therefore, it is supposed that the implementation of PFM based on a software developer that especially supports the solution of multifield problems should be more effective, simpler and more efficient than PFM implemented on a general finite element software. In this work, the authors aim to devise a simple and efficient implementation of phase-field model for the modelling of quasi-static and dynamic fracture in the general purpose commercial software developer, COMSOL Multiphysics. Notably only the tensile stress induced crack is accounted for crack evolution by using the decomposition of elastic strain energy. The width of the diffusive crack is controlled by a length-scale parameter. Equations that govern body motion and phase-field evolution are written into different modules in COMSOL, which are then coupled to a whole system to be solved. A staggered scheme is adopted to solve the coupled system and each module is solved sequentially during one time step. A number of 2D and 3D examples are tested to investigate the performance of the present implementation. Our simulations show good agreement with previous works, indicating the feasibility and validity of the COMSOL implementation of PFM.

A multi-mesh finite element method for phase-field based photonic band structure optimization

A novel finite element method with multiple meshes is proposed, which is applied to solve the phase-field models for photonic band structures optimization. In our approach, fine meshes are used for the phase field evolution, which allows fine resolution for shape representations. The coarse meshes are adopted for the finite element analysis of the state equation. Such a multi-mesh approach could save a considerable amount of computational costs. Numerical convergence is illustrated through comparisons between our computational results and benchmarks. The efficiency and robustness of the multi-mesh approach are also shown.

Numerical aspects of anisotropic failure in soft biological tissues favor energy-based criteria: A rate-dependent anisotropic crack phase-field model

A deeper understanding to predict fracture in soft biological tissues is of crucial importance to better guide and improve medical monitoring, planning of surgical interventions and risk assessment of diseases such as aortic dissection, aneurysms, atherosclerosis and tears in tendons and ligaments. In our previous contribution (Gültekin et al., 2016) we have addressed the rupture of aortic tissue by applying a holistic geometrical approach to fracture, namely the crack phase-field approach emanating from variational fracture mechanics and gradient damage theories. In the present study, the crack phase-field model is extended to capture anisotropic fracture using an anisotropic volume-specific crack surface function. In addition, the model is equipped with a rate-dependent formulation of the phase-field evolution. The continuum framework captures anisotropy, is thermodynamically consistent and based on finite strains. The resulting Euler–Lagrange equations are solved by an operator-splitting algorithm on the temporal side which is ensued by a Galerkin-type weak formulation on the spatial side. On the constitutive level, an invariant-based anisotropic material model accommodates the nonlinear elastic response of both the ground matrix and the collagenous components. Subsequently, the basis of extant anisotropic failure criteria are presented with an emphasis on energy-based, Tsai–Wu, Hill, and principal stress criteria. The predictions of the various failure criteria on the crack initiation, and the related crack propagation are studied using representative numerical examples, i.e. a homogeneous problem subjected to uniaxial and planar biaxial deformations is established to demonstrate the corresponding failure surfaces whereas uniaxial extension and peel tests of an anisotropic (hypothetical) tissue deal with the crack propagation with reference to the mentioned failure criteria. Results favor the energy-based criterion as a better candidate to reflect a stable and physically meaningful crack growth, particularly in complex three-dimensional geometries with a highly anisotropic texture at finite strains.

High-accuracy phase-field models for brittle fracture based on a new family of degradation functions

Phase-field approaches to fracture based on energy minimization principles have been rapidly gaining popularity in recent years, and are particularly well-suited for simulating crack initiation and growth in complex fracture networks. In the phase-field framework, the surface energy associated with crack formation is calculated by evaluating a functional defined in terms of a scalar order parameter and its gradients. These in turn describe the fractures in a diffuse sense following a prescribed regularization length scale. Imposing stationarity of the total energy leads to a coupled system of partial differential equations that enforce stress equilibrium and govern phase-field evolution. These equations are coupled through an energy degradation function that models the loss of stiffness in the bulk material as it undergoes damage. In the present work, we introduce a new parametric family of degradation functions aimed at increasing the accuracy of phase-field models in predicting critical loads associated with crack nucleation as well as the propagation of existing fractures. An additional goal is the preservation of linear elastic response in the bulk material prior to fracture. Through the analysis of several numerical examples, we demonstrate the superiority of the proposed family of functions to the classical quadratic degradation function that is used most often in the literature.

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 simulations of coupled microstructure solidification problems via the strong form particle difference method

This paper presents the development of a strong form-based collocation method called the particle difference method (PDM), capable of predicting the spatiotemporal evolution of polycrystalline material solidification through coupling of multi-phase and temperature fields. Cross coupled phase field evolution and heat transfer equations are discretized via the PDM to obtain the interface kinematics of polycrystalline boundary during solidification. A distinct feature of the PDM is its ability to represent derivative operators via a moving least-square approximation of the Taylor expansion through point-wise computations at collocation points. The method discretizes directly the strong forms using the pre-computed derivative operators at each collocation point and elegantly overcomes the topological difficulty in modeling intricate moving interfaces. To verify the efficacy of the PDM, numerical results are compared with those obtained from the conventional finite difference method for uniform and irregular distributions of the collocation points. The scalability of the parallelized PDM is tested by measuring its efficiency with increasing the number of processors. We also provide a solidification simulation with two ellipsoidal inclusions to demonstrate the capability of the PDM in complex moving interface problems with high curvature.

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.

Comparison of high order finite element and discontinuous Galerkin methods for phase field equations: Application to structural damage

Phase field equations are used to model a wide range of multiphase problems such as separation of fluids, solidification, viscous fingering, fracture and fatigue. A wide variety of methods to numerically solve phase field equations can be found in the literature. In particular, high order methods are an effective option when accuracy improvement is desired. In the first part of this work, we analyze the accuracy and computational efficiency of the high order finite element method (FEM) and discontinuous Galerkin (DG) method applied to the second-order Allen–Cahn (AC) and fourth-order Cahn–Hilliard (CH) equations. Several schemes for time integration are used for these equations. The explicit schemes are the forward Euler, classical fourth-order Runge–Kutta (RK4) and the strong stability preserving ten stages fourth-order Runge–Kutta (RKSSP-10,4) described in Gottlieb et al. (2011). The backward Euler and trapezoidal implicit methods are adopted in the full and semi implicit schemes, as proposed in Eyre (unpublished). Manufactured solutions for one dimensional problems are used in order to evaluate the errors and to compare the different numerical methods. By choosing an adequate discretization for AC equations resulting from the previous analysis of the first part of the work, in the second part, we propose a numerical semi implicit scheme to solve the damage and fracture model described in Boldrini et al. (2016). This procedure employs the FEM for spatial discretization, the Newmark method for time integration of the kinematics equation and the backward Euler for the damage phase field evolution. Finally, results for 2D benchmark tests are presented for the fracture phase field model and the convergence to a sharp crack for a small width of the damage phase field layer γ is verified.

Peridynamics damage model through phase field theory

We attempt a reformulation of the phase field theory in the framework of peridynamics (PD) to arrive at a continuum damage model. This obtains a better criterion for bond breaking in PD, marking a departure from the inherently ad-hoc bond-stretch-based or bond-energy-based conditions and thus allowing for the body to physically break into parts which a phase field model cannot by itself accomplish. Moreover, posed within the PD setup, the integral equation for the phase field eases the smoothness restrictions on the field variable. Taking advantages from both the worlds, the proposed scheme thus offers a better computational approach to problems involving cracks or discontinuities. Starting with Hamilton's principle, an equation of the Ginzburg-Landau type with dissipative correction is arrived at as a model for the phase field evolution. A constitutive correspondence route is followed to incorporate classical constitutive relations within our PD model. Numerical simulations of dynamic crack propagation (including branching) and the Kalthoff-Winkler experiment are also provided. To demonstrate how the model naturally prevents interpenetration, a mode II delamination simulation is presented. A brief discussion on the convergence of PD equations to the classical theory is provided in Appendix I.

A phase-field approach to model fracture of arterial walls: Theory and finite element analysis

This study uses a recently developed phase-field approach to model fracture of arterial walls with an emphasis on aortic tissues. We start by deriving the regularized crack surface to overcome complexities inherent in sharp crack discontinuities, thereby relaxing the acute crack surface topology into a diffusive one. In fact, the regularized crack surface possesses the property of Gamma-Convergence, i.e. the sharp crack topology is restored with a vanishing length-scale parameter. Next, we deal with the continuous formulation of the variational principle for the multi-field problem manifested through the deformation map and the crack phase-field at finite strains which leads to the Euler–Lagrange equations of the coupled problem. In particular, the coupled balance equations derived render the evolution of the crack phase-field and the balance of linear momentum. As an important aspect of the continuum formulation we consider an invariant-based anisotropic constitutive model which is additively decomposed into an isotropic part for the ground matrix and an exponential anisotropic part for the two families of collagen fibers embedded in the ground matrix. In addition we propose a novel energy-based anisotropic failure criterion which regulates the evolution of the crack phase-field. The coupled problem is solved using a one-pass operator-splitting algorithm composed of a mechanical predictor step (solved for the frozen crack phase-field parameter) and a crack evolution step (solved for the frozen deformation map); a history field governed by the failure criterion is successively updated. Subsequently, a conventional Galerkin procedure leads to the weak forms of the governing differential equations for the physical problem. Accordingly, we provide the discrete residual vectors and a corresponding linearization yields the element matrices for the two sub-problems. Finally, we demonstrate the numerical performance of the crack phase-field model by simulating uniaxial extension and simple shear fracture tests performed on specimens obtained from a human aneurysmatic thoracic aorta. Model parameters are obtained by fitting the set of novel experimental data to the predicted model response; the finite element results agree favorably with the experimental findings.