Phase Field Modeling Of Fracture Research Articles

An elastoplastic phase-field model for quasi-static fracture of nickel-based super-alloys

In the present study, an elastoplastic phase-field model of quasi-static fracture in ductile materials is proposed in the variational framework for J2 plasticity with isotropic hardening, which is suitable to describe the quasi-static behavior of metals as investigated in the performed experiments. These contributions include: (1) the free energy functions for coupling elastic response, plastic yielding and damage evolution are established. (2) The new elastic and plastic energy degradation functions are constructed to quantitatively describe the relationship between energy release and phase-field evolution of elastoplastic materials. (3) Damage evolution and plastic yielding criteria are derived. (4) From a numerical point of view, we derive the governing equations and the corresponding weak forms and the overall solution procedure for the phase-field model is given via the use of a return-mapping algorithm. This phase-field model was validated by a series of tensile experiments on Inconel 718 nickel-based super-alloys standard specimens. In order to compare the simulation results with the experimental results more comprehensively, the digital image correlation (DIC) technique is applied to experimentally investigate the specimen deformation information. In addition, to verify the potential of the model to capture complex cracks, we performed Nooru-Mohamed tests. The numerical simulation results are in good agreements with the results of previous experimental work.

The phase-field fracture model enriched by interpolation cover functions for brittle fracture problems

In this study, interpolation cover functions were introduced into the phase-field fracture model (PFM) to improve its convergence and efficiency. The traditional PFM has proven to be successful in simulating brittle fracture problems. However, one disadvantage is that it requires a relatively dense mesh. The PFM model enriched by interpolation cover functions (EPFM) was used for numerical fracture analysis under a relatively coarse mesh. The rank deficiency problem caused by the introduction of interpolation cover functions was resolved using the rank deficiency repair technique. Higher gradients were captured by introducing interpolation cover functions without changing the mesh structure. The numerical examples showed the improvement in convergence. The high efficiency of the proposed EPFM was demonstrated through a series of numerical examples. The numerical results of the EPFM and PFM under different length-scale parameters and meshes were also compared. The results showed that the proposed EPFM is an excellent alternative for evaluating engineering fracture problems.

Rate- and temperature-dependent ductile-to-brittle fracture transition: Experimental investigation and phase-field analysis for toffee

The mechanical behaviour of many materials, including polymers or natural materials, significantly depends on the rate of deformation. As a consequence, a rate-dependent ductile-to-brittle fracture transition may be observed. For toffee-like caramel, this effect is particularly pronounced. At room temperature, this confectionery may be extensively deformed at low strain rates, while it can behave highly brittle when the rate of deformation is raised. Likewise, the material behaviour does significantly depend on temperature, and even a slight cooling may cause a significant embrittlement.In this work, a thorough experimental investigation of the rate-dependent deformation and fracture behaviour is presented. In addition, the influence of temperature on the material response is studied. The experimental results form the basis for a phase-field modelling of fracture. In order to derive the governing equations of the model, an incremental variational principle is introduced. By means of the validated model, an analysis of the experimentally observed ductile-to-brittle fracture transition is performed. In particular, the coupling between the highly dissipative deformation behaviour of the bulk material and the rate-dependent fracture resistance is discussed.

A phase-field fracture model for fatigue behavior in fiber-reinforced composites

A new framework based on the phase field method is developed to model the fatigue fracture behavior of fiber-reinforced composite laminates. The growth path of matrix cracks can be simulated more accurately compared to the classical model. Further, anisotropic fatigue accumulation effects are considered in this model. Comparisons between simulations by the phase field model with uniform fatigue accumulation effects and the proposed method are encouraging. The new framework proposed can more naturally capture different types of crack growth in fiber- reinforced composites. The reliability and potential of this framework to model fatigue behavior for fiber-reinforced composite laminates are demonstrated by investigating several numerical examples. Moreover, the finite element implementation utilizes a simpler and robust approach via the ABAQUS user-defined subroutine UMAT and UMATHT, including the so called AT1, AT2 and PF-CZM models and two solution strategies.

A phase-field fatigue fracture model considering the thickness effect

Modern aero-engine turbine structures often have variable-thickness geometries (e.g., turbine structures with complex cooling systems) that pose a significant challenge to crack analysis. The phase-field fracture method is a numerical method that has recently received considerable attention. However, phase-field fatigue fracture models that consider the effect of thickness are still lacking. To address this problem, this paper proposes a modification of the fracture toughness in the phase-field fatigue model by introducing a relationship function between material fracture toughness and thickness (Gc-B curve). We use reference experimental data to fit the Gc-B curve of GH4169 and predict the fracture toughness of specimens with other thicknesses. The predictions of the proposed model agree well with the experimental results reported in the literature. This achievement contributes significantly to the structural safety design and life prediction of aero-engine components.

Modeling brittle crack propagation for varying critical load levels: a dynamic phase-field approach

AbstractBrittle materials are known for their violent and unpredictable cracking behavior. A behavior which is dictated by a combination of microscopical material defects and the competition between the potential energy of the system and the surface energy of the material. In this study, we present the implementation of a dynamic fracture phase-field model with a new crack driving force into a commercial finite element (FE) solver and examine its behavior using three different tension-compression splits. After validating the implementation, we use the model to investigate its predictive capacity on quasi-statically loaded L-shaped soda-lime glass specimens with varying critical load levels. The dynamic fracture phase-field model predicted similar crack propagation to what was found in the literature for quasi-static and dynamic validation cases. By varying the critical load level for the L-shaped soda-lime glass specimens using the new crack driving force, the model predicted a positive correlation between the initial crack propagation speed and the critical load level, similar to what was seen in the experiments. However, the predicted crack propagation speed decreased quicker than the experimental crack propagation speed. The tension-compression splits had an impact on the predicted crack propagation paths. Overall, the proposed crack driving force used in the dynamic fracture phase-field model seems to capture the relation between critical load and initial crack propagation speed and thus enables crack predictions for specimens of varying strength.

Fracture behaviors of nacre-like composites via phase-field fracture modeling

Nacre-inspired composites, composed of hard and soft phases, exhibit exceptional combinations of stiffness, strength, and fracture toughness. However, the fracture behaviors of these composites are significantly influenced by the geometric factors and material properties of their constituents. In this study, we systematically investigate the fracture behaviors of nacre-like composites by employing a phase-field fracture model considering the effects of oblique angle between loading and platelet directions, a critical factor often overlooked in previous studies, coupled with the effect of other geometric and material parameters, such as the aspect ratio, Young's modulus and fracture toughness ratio between hard and soft phase. Our findings highlight the significant influence of the oblique angle on the mechanical properties and fracture behaviors of these composites. Particularly, it is found the maximum Young’s modulus and strength are obtained when the direction of hard platelet is aligned with the tension direction, while the maximum toughness is achieved at a moderate oblique angle, especially in cases of large aspect ratios, small Young's modulus and fracture toughness ratios between the hard and soft phases. Besides, the maximum fracture toughness is more likely to be obtained at a small aspect ratio and large Young's modulus ratio between hard and soft phases. The insights gained from our study would offer valuable guidance for the design and practical application of nacre-mimetic composites.

Strengthening mechanism of steel fiber in UHPC: A new fracture phase field model

Strengthening mechanism of steel fiber in UHPC: A new fracture phase field model

An open-source moose implementation of phase-field modeling of fracture in functionally graded materials

Fracture analysis of functionally graded plates is a challenging problem in materials engineering due to the complex material behavior and the presence of cracks. In this study, we propose an efficient open-source MOOSE implementation of phase field fracture analysis of functionally graded. Automatically oriented exponential finite elements are used to discretize the model, which allows for efficient and accurate computation of the fracture behavior. The functionally graded material properties are modeled using a power law function, which captures the variation of material properties along the thickness direction of the plate. The phase field model is coupled with a finite element solver to simulate the mechanical response of the plate under loading. The proposed implementation is validated using several benchmark problems available in literature. The results show that the proposed MOOSE implementation provides accurate predictions of the crack propagation and fracture behavior of functionally graded plates.

Phase-field modeling of anisotropic crack propagation based on higher-order nonlocal operator theory

This paper presents a novel higher-order nonlocal operator theory for the phase-field modeling of brittle fracture in anisotropic materials. Incorporating higher order nonlocal operators can enhance the accuracy of the phase-field model by effectively capturing long-range interactions that hold significance in numerous materials. The reproducing kernel particle method is employed to derive a nonlocal differential operator to enhance computational stability and accuracy. Moreover, the proposed method eliminates the need for direct computation of derivatives of the modified kernel function, which avoids the calculation of moment matrix derivatives and improves computational efficiency. The phase-field modeling of polycrystalline materials, considering the anisotropic fracture resistance of each grain, is implemented using this numerical framework. The present method is able to capture different scenarios intergranular and transgranular crack propagation patterns in polycrystalline materials. The proposed method involves a detailed representation of the complex process of crack initiation and propagation in 2D and 3D models of polycrystalline materials.

Adaptive meshfree method for fourth-order phase-field model of fracture using consistent integration schemes

An adaptive fourth-order phase-field model is proposed using consistent element-free Galerkin method (EFG). Convenient construction of high-order interpolation functions in EFG method is fully utilized. Requirement of C1 continuity of shape functions is satisfied in fourth-order phase-field model. Two different consistent integration schemes with high efficiency and high accuracy, namely quadratically consistent one-point integration scheme and three-point integration scheme, are employed to evaluate the fourth-order phase-field equation and displacement equation, respectively. To reduce computational cost, the adaptive strategy of local background mesh refinement is established on basis of strain energy history and phase field. Background mesh is refined via the insertion of nodes at the midpoints of each side of the integration element. Comparing with EFG method, computational accuracy and efficiency are enhanced by the proposed method. Load-displacement response, the smoothness of the resulting stress and crack paths are predicted well.

Evaluating tensile strength of cement paste using multiscale modeling and in-situ splitting tests with micro-CT

This study investigated a strategy to simulate splitting tests to evaluate the tensile strength of cement paste within a multiscale framework, and validated the approach with in-situ micro-CT measurements. Phase-field fracture model was used to simulate the splitting test using virtual microstructures reconstructed from the macroscale micro-CT measurements. Among the modeling parameters, the range of the diffusive crack width, which is an internal parameter for phase-field fracture model, was further investigated. The cracks propagated through the weaker solid phase, and the predicted strengths and standard deviations were comparable to those obtained from the experiments. It was found that the diffusive crack width should be smaller than the characteristic length of the weaker phase to predict the crack pattern accurately. These findings confirm the effectiveness of the proposed multiscale modeling approach in predicting the tensile strength of cement paste.

Electro-mechanical actuation modulates fracture performance of soft dielectric elastomers

Soft dielectric elastomers respond to electric stimuli by undergoing large deformations and changes in their material properties. The actuation with deformable electrodes attached to the material originates Coulomb and dipole forces that convert the electric field into a mechanical response. Applications at large deformations can entail crack onset and propagation. Within this context, the response of a soft polymer to an applied electric field may serve to influence the fracture behavior of such materials, potentially enhancing it. Here we explore the fracture performance of an ultra-soft dielectric elastomer. To do so, we conduct tensile tests while applying electrical actuation on samples with pre-cuts. Additionally, we examine the elastomer filled with piezoelectric BaTiO3 particles to ameliorate the fracture performance beyond the limits observed in the unfilled material. In conjunction with the experiments, we employ a bespoke fracture phase-field model to analyze the stress triaxiality near the crack tip. The results indicate that the electric actuation induces beneficial crack tip blunting and stress de-concentration, enhancing the fracture toughness up to a 125% and delaying crack propagation. Our work provides a route for applications of soft dielectric elastomers that require improved fracture properties or, more broadly, the modulation of fracture behavior.

A phase field fracture model for ultra-thin micro-/nano-films with surface effects

Surface effects usually remarkably affect the mechanical response of ultra-thin micro-/nano-structures. However, the mechanisms of surface effects on the fracture characteristics of ultra-thin films are still not fully understood. To this end, this paper develops a modeling framework to investigate the fracture of ultra-thin films at microscales or below. Such a framework couples the Gurtin–Murdoch theory with a phase-field fracture model, in which the former is adopted to introduce the surface effects, i.e., the surface residual stress and surface elasticity of a thin film, and the latter is able to model crack evolution without requiring predefined crack paths or any criteria. Furthermore, a novel crack driving force is introduced, which encompasses the tensile components of both bulk elastic energy and surface elastic energy. Several numerical examples including the biaxial tension test as well as the single-edge notched tension/shear test are performed. The simulation results indicate that the surface strain energy plays a major role in the total elastic strain energy of an ultra-thin film when its thickness is at a micro level, thus demonstrating the significance of surface effects. Moreover, the mode-I fracture test shows that the surface elasticity and surface residual stress have a remarkable influence on the displacement at failure, while for the mode-II fracture test, the surface residual stress significantly influences the fracture characteristics such as the crack path and failure displacement. The developed model paves the way for revealing the fracture mechanisms of ultra-thin micro-/nano-films and conducting their safety assessment.

Phase-field description of fracture in NiTi single crystals

A phase-field model for thermomechanically-induced fracture in NiTi at the single crystal level, i.e., fracture under loading paths that may take advantage of either of the functional properties of NiTi–superelasticity or shape memory effect–, is presented, formulated within the kinematically linear regime. The model accounts for reversible phase transformation from austenite to martensite habit plane variants and plastic deformation in the austenite phase. Transformation-induced plastic deformation is viewed as a mechanism for accommodation of the local deformation incompatibility at the austenite–martensite interfaces and is accounted for by introducing an interaction term in the free energy derived based on the Mori–Tanaka and Kröner micromechanical assumptions and the hypothesis of martensite instantaneous growth within austenite. Based on experimental observations suggesting that NiTi fractures in a stress-controlled manner, damage is assumed to be driven by the elastic energy, i.e., phase transformation and plastic deformation are assumed to contribute in crack formation and growth indirectly through stress redistribution. The model is restricted to quasistatic mechanical loading (no latent heat effects), thermal loading sufficiently slow with respect to the time rate of heat transfer by conduction (no thermal gradients), and a temperature range below Md, which is the temperature above which the austenite phase is stable, i.e., stress-induced martensitic transformation is suppressed. The numerical implementation of the model is based on an efficient scheme of viscous regularization in both phase transformation and plastic deformation, an explicit numerical integration via a tangent modulus method, and a staggered scheme for the coupling of the unknown fields. The model is shown able to capture transformation-induced toughening, i.e., stable crack advance attributed to the shielding effect of inelastic deformation left in the wake of the growing crack under nominal isothermal loading, actuation-induced fracture under a constant bias load, and crystallographic dependence on crack pattern.

Two different micro-polar phase-field models for brittle fracture and their open-source finite element implementation

Realizing the limitations of classical phase-field fracture models and the need for the development of non-local phase-field models that can capture the experimentally observed size effects in brittle and quasi-brittle materials, we propose thermodynamically consistent volumetric-deviatoric and spectral decomposition-based phase-field models (PFMs) for brittle fracture in micro-polar continua. For developing the proposed models, we have considered the micro-rotation as an additional kinematic descriptor, and have derived the balance equations by invoking the virtual power principle. On satisfying the thermodynamic laws, we have determined the constitutive relations for the thermodynamic fluxes and demonstrated that any dissipative effect can easily be incorporated into the proposed models. We have provided an open-source finite element (FE) implementation of the micro-polar PFMs using Gridap in Julia. The implementation includes various benchmark problems to illustrate how the responses in a specimen change with the variations in micro-polar material parameters. Through a series of numerical examples with different values of micro-polar material parameters, we have showcased both the capabilities and limitations of the proposed volumetric-deviatoric and spectral decomposition-based micro-polar PFMs.

Phase field modeling of fatigue crack growth retardation under single cycle overloads

A zone-based crack retardation model that responds to single cycle overloads is applied to a phase field fatigue framework. Fatigue crack growth is incorporated through the degradation of fracture toughness in the phase field fracture model where a representative loading strategy is followed instead of explicit cyclic loading. The model is demonstrated to capture Paris-Erdogan law type behavior. Crack growth retardation following an overload cycle is simulated through a zone ahead of the crack tip taking inspiration from existing plastic zone-based retardation models. The retardation zone is described through a strain energy density limit, with the overload ratio controlling the extent to which the fatigue damage accumulation rate is slowed inside the retardation zone. The model is implemented utilizing user subroutines in Abaqus with coupled-temperature displacement elements where temperature acts as a stand in for the phase field parameter. The model is found capable of reproducing experimental levels of fatigue life gains and reduction in crack growth rate for compact tension and center-cracked tension panel specimens when various overload levels are applied. Furthermore, the model is demonstrated to be able to capture complex crack paths with great accuracy.

An adaptive multi-patch isogeometric phase-field model for dynamic brittle fracture

This paper presents an adaptive multi-patch isogeometric phase-field model based on Nitsche's method for simulating dynamic brittle fracture. Complex structures are accurately represented with multiple patches, and Nitsche's method is used to link two adjacent patches to ensure the compatibility and continuity of physical quantities (such as displacements, stresses, phase-field variables) on the coupling edge. LR NURBS with the local refinement ability are used for spatial discretization, while a generalized-α method is used for temporal discretization. The staggered algorithm and hybrid PFM formulation are adopted to enhance the solution efficiency of multi-field coupled system. In addition, the local refinement is executed utilising the phase-field threshold. Several dynamic fracture examples are used to validate the proposed approach's accuracy and effectiveness. The proposed approach can effectively model dynamic fracture propagation, and numerical results demonstrate that the adaptive analysis may significantly lower the computing cost.

Study of mixed-mode fracture in functionally graded material using an adaptive phase-field fracture model

This work systematically investigates the mixed-mode fracture propagation in functionally graded materials. The crack trajectory and load–displacement responses were effectively simulated using the mixed-mode phase field approach in combination with an adaptive mesh refinement technique. The mesh refinement is carried out using quadtree decompositions, and the elements with hanging nodes are handled as n−sided polygons. The adaptive implementation is validated by solving benchmark problems from the literature. The adaptive technique demonstrates greater computational efficiency by significantly reducing the number of elements without compromising the accuracy of the solution. This study provides insight into how fracture propagation behavior is influenced by gradient profile, material gradation, crack location, energy release rate, and displacement configuration in uni-directional and bi-directional functionally graded material. Remarkably, it is found that the peak load on the load–displacement graph changes noticeably, and the fracture path remains constant despite variations in the critical energy release rate. Notably, the results show that materials with lower fracture toughness have an early failure when studying a center fractured plate. This thorough examination deepens our understanding of the propagation of mixed-mode cracks and provides insight into improving the analysis and design methodologies for functionally graded materials.

A robust tension-compression asymmetric phase-field fracture model for describing PBX cracking under complex stress states

ABSTRACT The crack behaviors under complex stress states are very important for the safety of polymer-bonded explosives (PBXs) under accidental stimulations, but their accurate description is a challenge. Due to the advances of tracking discontinuities and multi-fields coupling, the phase-field model for complex fracture phenomena is attracting significant interest recently. Conventional phase-field fracture models are tension-compression symmetric or based on volumetric-deviatoric strain energy split, and these conventional phase-field models may lead to unrealistic fracture patterns, which hinders its further application in PBX fracture simulations. In this work, we present an extended, tension-compression asymmetric phase-field fracture model for PBXs, which distinguishes the contributions of tensile and compressive stresses to damage driving energy, and couples the mechanism of mechanical degradation and energy-driving cracking diffusion. We implemented our improved phase-field fracture model into finite element calculations and compared the simulation results with the conventional tension-compression symmetric phase-field fracture model and volumetric-deviatoric strain energy split phase-field fracture model by simulating PBX specimens under static loadings. The results show that our model not only accurately depicts the tensile and compressive cracks, but also describes compression-assisted cracking while suppressing unrealistic damage nucleation caused by small amplitudes of local compressive stresses, making it a very efficient way of describing PBX cracking under complex stress states. This new model is both mathematically and physically concise, and convenient for numerical implementation. Moreover, the novel model can be naturally extended to simulate shock-induced dynamic and/or coupled fracture of PBXs because of its feasibilities for dynamic extension and multi-field coupling.