Adaptive Phase Field Model Research Articles

On the use of scaled boundary shape functions in adaptive phase field modeling of brittle fracture

On the use of scaled boundary shape functions in adaptive phase field modeling of brittle fracture

Dynamic crack growth in orthotropic brittle materials using an adaptive phase-field modeling with variable-node elements

In this paper, crack growth in orthotropic materials subjected to dynamic loading is numerically studied using an adaptive phase-field method. The study starts with a coarse structured mesh and the adaptive refinement strategy based on a user-defined threshold on the phase-field variable is proposed for computational efficiency, and variable-node elements are employed to treat the hanging nodes as a result of local adaptive refinement. The Hughes-Hilbert-Taylor (HHT) time integration scheme is adopted for temporal discretization. The directionality of orthotropic materials is represented by a penalized second-order structural matrix, which is incorporated in the crack face energy density. Through numerical examples, the influence of the material orientation on the dynamic crack growth in orthotropic materials is studied and the reliability of the proposed framework is validated.

Low-cycle fatigue crack growth in brittle materials: Adaptive phase-field modeling with variable-node elements

This work presents an adaptive phase field framework for the simulation of crack nucleation and propagation in brittle materials subject to low-cycle loading. We introduce a fatigue history strain parameter within the phase-field framework to capture the fatigue effect. The resulting coupled differential equations are solved using a staggered iteration scheme. In order to improve the computational efficiency of the phase-field model, an adaptive refinement scheme is introduced. This scheme utilizes an artificial threshold that takes into consideration of both phase-field variables and cumulative fatigue history field variables to determine the elements to be refined. The handling of the hanging nodes resulting from mesh refinement is accomplished through the application of the variable-node element technique, offering a flexible and efficient mesh integration technique. We validate our proposed numerical scheme through five representative numerical examples: single-edge cracked specimen, compact tension specimen, double-edge cracked panel, plate with a hole and plate with multi-cracks and hole. The results demonstrate that the adaptive mesh refinement scheme significantly reduces computational costs without compromising the accuracy of the numerical predictions.

Implementation of the adaptive phase-field method with variable-node elements for cohesive fracture

In this paper, an adaptive phase-field model based on variable-node elements and error-indicator is presented to predict cohesive fracture evolution. The phase-field cohesive-zone model is characterized by the parameterized energetic degradation and crack geometric functions. A staggered iteration scheme is used to solve the coupled non-linear system of displacement field and phase field. The adaptive local refinement algorithm facilitated with error-indicator and variable-node elements is developed to improve computational efficiency. To determine the adaptive local refinement regions, we apply the phase-field threshold to characterize the crack regions and a novel history field threshold to denote crack tips. The variable-node elements are acted as transition elements to flexibly yet simply link the fine and coarse meshes. In addition, a fast method to generate the variable-node elements is given. Several representative numerical examples are studied to demonstrate the performance and accuracy of the proposed adaptive phase-field model, consisting of a mixed-model fracture of L-shape panel, the wedge-splitting test, the classic three point bending experiment and a plate with a hole and multi-cracks under tensile/shear loadings. The computer codes can be accessed at: https://github.com/hhuztc/Adaptive-PFM.git.

Realization of adaptive mesh refinement for phase-field model of thermal fracture within the FEniCS framework

This paper presents an adaptive phase field model for simulating fracture due to coupled interactions (such as thermal quenching and hot cracking in additive manufacturing). The proposed model is implemented in an open-source finite element framework, FEniCS. The model considers spatial variations of the fracture toughness and differential coefficients of thermal shrinkage. Several paradigmatic case studies are addressed to demonstrate the potential of the proposed modeling framework. Specifically, we (a) benchmark our crack predictions for mechanical and thermal boundary condition interactions with the results from alternative numerical methods, (b) accurately reproduce experimentally observed complex crack trajectories due to thermal quenching and hot cracking in additive manufacturing, and (c) demonstrate the ease of extending of the proposed framework to thermal cracking problems in three dimensions. The current implementation provides the basic for an efficient framework for fracture problems due to multi-physical interactions for practical engineers with less programming expertise.

Adaptive phase-field modeling of fracture propagation in bi-layered materials

We study fracture propagation in bi-layered materials using an adaptive phase-field method. The method combines a phase-field representation of fracture with spatial adaptivity to provide mesh refinement in the smeared zone of damage, where higher accuracy is needed to resolve the solution. We place a particular emphasis on studying the competition between crack arrest, deflection, and penetration at the interface of two dissimilar materials. To explore this competition, we perform tension tests on various models with different combinations of material and geometric properties. We begin by validating the adaptive phase-field model through a uniaxial tension test. The material mismatch and locations of the initial damage for this example are chosen such that the fracture propagation behavior is identical to that in a homogeneous material. Next, we independently vary the mismatches in elastic stiffness and critical energy release rate to study their influence on crack growth. Our observations show that both mismatches contribute to an effective toughening of the model, but in different ways. Critical energy release rate mismatch causes crack deflection along the interfaces, while elastic stiffness mismatch results in retardation of crack growth. We also note that for a specific combination of material mismatch and location of initial damage, crack nucleation in the adjacent material is preferred over propagation from the existing notch. Finally, through models with inclined interfaces, we demonstrate the effect of the relative orientation between the approaching crack and the material interface on crack penetration, deflection, and nucleation. The numerical results from this study provide valuable insight into the various failure mechanisms in bi-layered materials.

Crack growth in anisotropic brittle and polycrystalline materials by adaptive phase field model using variable-node elements

Crack growth in anisotropic brittle and polycrystalline materials by adaptive phase field model using variable-node elements

An efficient adaptive length scale insensitive phase-field model for three-dimensional fracture of solids using trilinear multi-node elements

As a diffused fracture theory, phase-field models can seamlessly simulate complex crack patterns such as extending, branching, and merging. Despite the success of phase-field models, there are two issues in previous methods of three-dimensional (3-D) fracture. Firstly, the nonlinear governing equations lead to the huge computational costs, which hinder the application of phase-field models in 3-D problems. Secondly, these models, which are mostly developed based on a simple quadratic degradation function, provide numerical solutions that are sensitive to a length scale. Hence, this work addresses an efficient adaptive phase-field model with the aid of trilinear multi-node elements. The order of the elements remains constant with the increase of the number of nodes. As the mesh size and length scale significantly influence the numerical precision, a robust adaptive criterion is established in which the element refinement is controlled by both internal length scale and phase-field. According to the criterion, an expected mesh density in the failure domain can be obtained even for nonuniform initial mesh. Besides, being able to extend the phase-field regularized cohesive zone model, the adaptive model provides length scale insensitive responses for both crack path and peak load. The failure of brittle and quasi-brittle materials in three-dimensional conditions, including simple and mixed-mode fracture, can be simulated by the proposed model. Several benchmark examples are analyzed to show the efficiency and accuracy of the trilinear element-based adaptive phase-field model (TAPFM), and the results are compared with the standard phase-field model as well as experimental data.

Adaptive isogeometric analysis–based phase-field modeling of interfacial fracture in piezoelectric composites

This paper implements an adaptive phase-field model to investigate the interfacial fracture in transversely isotropic piezoelectric composites under different electromechanical loadings using an isogeometric framework based on polynomial splines over hierarchical T-meshes (PHT-splines). The model introduces the interface and crack phase-field parameters to regularize the sharp interface and crack topologies, respectively and the regularized energy terms are incorporated corresponding to crack and interface evading the need for line integration along them. It also modifies the energy functional to scrutinize the interaction between interfacial damage and crack propagation in piezocomposites and thus is capable of capturing the fracture patterns including interfacial debonding, matrix cracking and their interactions under different electric fields. An adaptive mesh refinement scheme using PHT-splines is implemented to efficiently resolve for interface and crack phase-fields and track the complex crack propagation paths without any ad hoc criteria. The efficacy and robustness of the proposed model are demonstrated using a set of examples simulating the electromechanical fracture in piezoelectric composites. Moreover, different fracture mechanisms in composite materials including the crack nucleation, interfacial debonding, matrix cracking, crack propagation and coalescence are found to be precisely captured using the present modeling technique.

Quasi-static thermoelastic fracture: Adaptive phase-field modeling with variable-node elements

Although robust in handling different fracture processes such as nucleation, branching and coalescence, the phase-field method (PFM) is computationally very expensive because it requires extremely fine meshes to resolve the necessary physics. This paper presents a hybrid adaptive PFM discretized by using finite element method to model quasi-static fracture of thermoelastic solids and quenching. Based on a user-defined threshold on the phase field variable, the computational domain is local refined. The incompatibility between the meshes due to local refinement is directly handled by the variable-node elements, without the special handling of hanging-nodes. The coupled phase-field thermo-elastic equations are solved using the hybrid approach combined with a staggered solution scheme, and the robustness of the proposed framework in terms of capturing the crack morphology is demonstrated with several standard benchmark problems.

An adaptive phase-field model with variable-node elements for fracture of hyperelastic materials at large deformations

In this work, an adaptive phase-field method is proposed for modeling fracture of hyperelastic materials at large deformations. The adaptive mesh refinement is facilitated by the variable-node elements, which are flexible to act as transition elements in the employed quadtree mesh. To control the adaptive process, we propose a combined phase-field and energetic mesh refinement criterion, where the energetic part exploits the strain energy threshold of the AT1 phase-field model which is used in this work. Both the compressible and incompressible neo-Hookean models are taken into account, and the latter is enforced by the plane stress condition to simplify the implementation. Several representative examples are studied to verify the accuracy and efficiency of the proposed adaptive phase-field method, in comparison to the available numerical and experimental reference data as well as the fixed locally pre-refined mesh. The simulated results show that the energetic part of the mesh refinement criterion can effectively prevent the delayed damage evolution when the phase-field initiates. Finally, the fracture process of a hyperelastic composite with inclusions is simulated to demonstrate the capacity of the proposed method for reproducing complex failure phenomena at large deformations.

Extension of the spatially adaptive phase-field model to various forms of fracture

The phase field approach has proved to be efficient and has received ample attention amongst the available techniques to model fracture. However, high computational cost still imposes substantial difficulties in the phase-field simulation of fractures. This contribution is based on a recently proposed variational approach for spatial adaptivity in a phase-field model of fracture. The main idea is to consider the regularisation length ϵ as a space-dependent variable in the argument of the energy functional. We extend this now by implementing a strain energy split to ensure that only the tensile energy drives the crack propagation. The displacement, phase field, and optimal regularisation length are then determined locally by minimisation of the modified energy functional. Subsequently, the computed optimal regularisation length is used to refine the mesh size locally. The resultant solution procedure is implemented in the finite element library FEniCS. Numerical investigations on selected examples of different fracture modes demonstrate that the spatially adaptive phase field model has a comparable convergence rate, but a subjacent energy convergence curve resulting in significant computational savings. Moreover, it also computes the peak force more accurately illustrating its potential for usage in practical applications.

Adaptive isogeometric analysis–based phase-field modeling of brittle electromechanical fracture in piezoceramics

The investigation of brittle fracture in piezoceramics under complex electromechanical loading is critical for their durable design and optimal utilization. Phase-field modeling offers a convenient and effective strategy to tackle three-dimensional (3D) fracture problems through the regularization of sharp crack topologies. This paper aims to develop an adaptive phase-field model to study electromechanical fracture in piezoceramics via an isogeometric formulation based on polynomial splines over hierarchical T-meshes (PHT-splines). In particular, we (i) consider the evolution of the crack phase-field within the framework of coupled electromechanical constitutive relationships, (ii) implement PHT-splines to make adaptive refinement computationally efficient and overcome the limitation of nonuniform rational B-splines-based isogeometric formulations, (iii) benchmark our findings with experiments and other numerical studies, and (iv) capture complex crack propagation patterns including deflection and twisting under different complex electromechanical loading conditions in 2D and 3D cracked piezoceramics. The computational efficiency of the implemented phase-field model in cracked piezoceramics is improved through facilitating the adaptive mesh refinement during crack propagation. The proposed scheme lays down the foundation for modelling the diffusive electromechanical fracture in cracked piezoceramics.

A spatially adaptive phase-field model of fracture

Phase-field models of fracture introduce smeared cracks of width commensurate with a regularisation length parameter ε and obeying a minimum energy principle. Mesh adaptivity naturally suggests itself as a means of supplying spatial resolution where needed while simultaneously keeping the computational size of the model as small as possible. Here, a variational-based spatial adaptivity is proposed for a phase-field model of fracture.An extension of the conventional phase-field model is achieved by allowing spatial variation of the regularisation length ε in the energy functional. Similar to the displacement and phase fields, the optimal regularisation length is obtained by minimising the energy functional. This extended phase-field model serves as the foundation for an adaptive mesh refinement strategy, in which the mesh size is determined locally by the optimal regularisation length. The resulting solution procedure is implemented in the framework of the finite element library FEniCS. According to the selected numerical experiment, the spatially adaptive phase-field model converges marginally faster than the conventional phase-field model but with a vastly superior constant, resulting in significant computational savings.

An adaptive phase-field model based on bilinear elements for tensile-compressive-shear fracture

An efficient adaptive phase-field method based on bilinear elements for tensile-compressive-shear fracture is developed. On the one hand, refined meshes are needed near the crack path to obtain accurate results, and the computational efficiency is very low if a prior mesh refinement is applied, especially when the crack paths are unknown. On the other hand, conventional phase-field models are not applicable to tensile-compressive-shear fracture under complex stress states at present. In this paper, an adaptive scheme is proposed to improve the computational efficiency, in which a bilinear multi-node element is adopted to avoid using high order quadrature and shape functions when new nodes are inserted in elements, and a new multi-node triangular element is outlined to expand the scope of application of the proposed adaptive method. Another important aspect of the contribution is the development of the phase-field model for tensile-compressive-shear fracture under complex stress states, in which a universal fracture criterion is embedded. To determine the optimal parameters in the adaptive scheme and demonstrate the advantages of the proposed bilinear adaptive method, a series of numerical examples are performed for sensitivity analysis. Comparison with experimental results is also conducted to validate the proposed phase-field model for tensile-compressive-shear fracture under complex stress states.

Adaptive higher-order phase-field modeling of anisotropic brittle fracture in 3D polycrystalline materials

Due to its ability to simulate complex microstructure evolution, the phase-field modeling has been extensively developed to investigate brittle fracture in recent years. However, low computational efficiency still imposes substantial difficulties in the development of phase-field models. In this work, we develop a novel adaptive phase-field approach based on the isogeometric meshfree collocation method (IMCM) to simulate the crack propagation in 2D and 3D polycrystalline materials. The concept of IMCM is based upon the correspondence between the isogeometric collocation and reproducing kernel meshfree method to facilitate a robust mesh adaptivity in isogeometric collocation. The strong form collocation formulation further enhances the computational efficiency of phase-field modeling by reducing the number of point evaluations. The present numerical framework is utilized for the adaptive phase-field modeling which introduces the anisotropy of fracture resistance for each grain in polycrystals. Furthermore, the discrete displacement and phase-field equations are generalized to enable the calculation of both second- and fourth-order gradients, which are required to solve the phase-field models using IMCM. The smoothness and higher-order continuity of IMCM enable the fourth-order phase-field equation to be solved directly without splitting it into two second-order differential equations. The fourth-order model can capture the crack surface accurately with fewer nodes than the second-order model. Several numerical examples of isotropic and anisotropic brittle fracture in polycrystalline materials are investigated to demonstrate the effectiveness and robustness of the proposed approach.

Adaptive phase-field modeling of brittle fracture using the scaled boundary finite element method

In this work, we propose an adaptive phase field method (PFM) to simulate quasi-static brittle fracture problems. The phase field equations are solved using the scaled boundary finite element method (SBFEM). The adaptive refinement strategy is based on an error indicator evaluated directly from the solutions of the SBFEM without any need for stress recovery techniques. Quadtree meshes are adapted to perform mesh refinement. The polygons with hanging nodes in the quadtree decomposition are treated as n−sided polygons within the framework of the SBFEM and do not require any special treatment in contrast to the conventional finite element method. Several benchmark problems are used to demonstrate the robustness and the efficacy of the proposed technique. The adaptive refinement strategy reduces the mesh burden when adopting the PFM to model fracture. Numerical results show an improvement in the computational efficiency in terms of the number of elements required in the standard PFM without compromising the accuracy of the solution.

Understanding the facet formation mechanisms of Si thin-film solidification through three-dimensional phase-field modeling

Adaptive phase field modeling is used in order to model the formation mechanism of a silicon faceted interface in three dimensions. We investigate the faceting condition for equilibrium shapes and dynamic situations. In this study, we propose a new anisotropic function of surface energy for the phase-field simulations in three-dimension, and negative stiffness is further considered. The morphological evolutions are presented and compare well with experimental findings. The growth mechanism is further discussed.

Adaptive Phase-Field Modeling of Anisotropic Wetting with Line Tension at the Triple Junction.

Line tension could affect the contact angle at triple junction, especially in micro- to nanoscale wetting. We have developed an adaptive phase-field model to consider the line tension quantitatively. This model is coupled to the smoothed boundary method for treating the contact line with the solid phase, while the volume constraint is imposed. Our calculated contact angles are in good agreement with the modified Young's equation. Further examples are illustrated for the anisotropic wetting on hydrophilic/hydrophobic stripes and rectangular grooves.

Three-dimensional phase field modeling of silicon thin-film growth during directional solidification: Facet formation and grain competition

Adaptive phase field modeling is applied to simulate a directional solidification for silicon in three dimensions. Several parameters are considered for the highly anisotropic interfacial energy and kinetic coefficient of silicon. The morphological evolution and the thermal field near the interface are presented and compared with experiments. The mechanisms for facet formation and grain competition are further discussed.