Phase Field Method For Fracture Research Articles

Study of adhesive wear mechanisms in asperity junctions based on phase field fracture method

Adhesive wear that is mostly induced by adhesion between asperities of rough surfaces in contact is one of the most common forms of wear of materials. However, the adhesive wear mechanisms are still not fully understood. In this work, the phase field fracture method that does not require any assumption on the crack path is used to explore the debris formation mechanism during asperity separation in adhesive wear. It is found that for ideal materials without defects the size of wear particles depends on the contact length. A small contact length leads to fracture at the junction and the formation of small wear particles. On the contrary, a large contact length will cause fracture in the bulk, resulting in large wear particles. Moreover, the angle of crack propagation decreases with increasing base angle of asperity and the ratio of the asperity height to the base edge width. The variation of fracture angle is mainly related to the ratio of asperity height to the base edge width that will influence the angle of the interaction force between asperities in the wear process. For materials with initial hole defect, the formation mechanism of wear particles will be little affected when the hole is rather small or far away from the junction. However, as the hole diameter further increases, the wear particles will change from large particles to small ones. And for materials with initial crack defect, the crack propagation always starts from the initial crack during adhesive wear. Moreover, for adhesive wear of materials with an initial crack at the bottom of asperity, fracture angle in the bulk is significantly positively correlated with the contact length and large wear particles are easy to form. However, during adhesive wear of asperity with the initial crack inside the bulk of asperity, small particles are more likely to form. This work contributes to further understanding of adhesive wear process.

Experimentally calibrated viscoelastic phase-field fracture method of thermoplastic resins

In the past few years, there has been considerable emphasis on investigating fracture behavior in composite materials using the phase-field fracture (PFF) method. However, accurately characterizing the viscoelasticity of the matrix necessitates employing the viscoelastic PFF method, and there remains an ongoing debate regarding the precise determination of the viscous driving force introduced in this methodology. This study aims to propose a feasible calibration method for quantifying the viscous driving force. To achieve this goal, benchmark experiments utilizing Polyetheretherketone (PEEK), a typical thermoplastic resin, are carried out by designing uniaxial tension tests, three-point bending tests, as well as compression tests at various strain rates. Through employing the viscoelastic PFF method, we discuss the effect of strain rate and viscous driving force on mechanical response. Our findings reveal a non-linear relationship between the influence of viscous driving force and strain rate with an initial increase followed by a subsequent decrease. By combining simulation results with experimental data analysis, the viscous driving force value is determined. The results demonstrate that our proposed approach is reliable for quantifying the viscous driving force in the viscoelastic PFF method, laying a foundation for investigating the mechanical response of composite materials.

Multiphysics phase-field modeling for thermal cracking and permeability evolution in oil shale matrix during in-situ conversion process

In-situ conversion process (ICP) is a promising recovery technology for economically effective exploitation of unmatured oil shale. High temperature treatment during ICP facilitates the chemical conversion of kerogen content in oil shale, while markedly improving the accessibility of pyrolyzed fluid through thermally-induced microcracks. Although extensively studied experimentally, the microstructure development in shale is not fully understood due to the lack of consideration for multiphysics effects during ICP at the mineral scale. This paper presents a numerical study on thermal cracking and permeability evolution in oil shale matrix during in-situ conversion process. To this end, a multiphysics phase-field model for investigating thermo-mechanical response, chemical reaction, and seepage behavior is developed, while arbitrary crack growth in heterogeneous shale matrix is naturally predicted using the phase-field method for fracture. Implementing the proposed numerical model, thermal cracking of heterogeneous granodiorite is first simulated, from which the numerical outcome regarding crack morphology is reasonably consistent with experimental data. Permeability evolution of oil shale matrix is found to be attributed to early-stage thermal crack propagation and late-stage kerogen decomposition by correlating multiple variables. Anisotropic permeability is also observed and investigated by examining crack morphology, pore-space interconnectivity and fluid permeation. Further analysis reveals that the shale matrix microstructure, in-situ stress and heating temperatures play vital roles in influencing thermal cracking and permeation behaviors of the shale matrix. The results provide a unique perspective to understanding thermal cracking and permeability augmentation of heterogeneous rocks.

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.

Thermal Shock Fracture of Functionally Graded Materials Based on the Phase-Field Fracture Method

Thermal Shock Fracture of Functionally Graded Materials Based on the Phase-Field Fracture Method

Numerical Study on Fracture Behavior of Printed Circuit Heat Exchanger Core Based on Elasto-Plastic Phase Field Method

Abstract Printed circuit heat exchanger (PCHE) is a type of compact heat exchanger with high thermal efficiency, more compact, and suitable for high pressure and high temperature conditions. The safety assessment of PCHE is very important, especially for PCHE core. Because PHCE core is different from the PCHE headers which are typical pressure vessel components. However, there is no standard design method to ensure the structural integrity for PCHE core. Some failure modes of PCHE core are studied in recent years, but seldom researches have been conducted concerning the fracture behavior of PCHE core. In addition, the phase field method for fracture has developed rapidly in recent years with its advantages in predicting crack initiation, bifurcation, and merging. In this work, the fracture behavior of PCHE core with and without defects is investigated based on elasto-plastic phase field method. The model parameter of Ni-based Alloy for elasto-plastic phase field method is first calibrated by comparing the numerical results with tensile test data of diffusion-bonded PHCE core specimen. Then, the influence of loading directions, defect locations, and defect sizes on the fracture behavior of PCHE core with semicircular and semicircular-rectangular channels are analyzed. Some conclusions are discussed.

On formulations for modeling pressurized cracks within phase-field methods for fracture

Over the past few decades, the phase-field method for fracture has seen widespread appeal due to the many benefits associated with its ability to regularize a sharp crack geometry. Along the way, several different models for including the effects of pressure loads on the crack faces have been developed. This work investigates the performance of these models and compares them to a relatively new formulation for incorporating crack-face pressure loads. It is shown how the new formulation can be obtained either by modifying the trial space in the traditional variational principle or by postulating a new functional that is dependent on the rates of the primary variables. The key differences between the new formulation and existing models for pressurized cracks in a phase-field setting are highlighted. Model-based simulations developed with discretized versions of the new formulation and existing models are then used to illustrate the advantages and differences. In order to analyze the results, a domain form of the J-integral is developed for diffuse cracks subjected to pressure loads. Results are presented for a one-dimensional cohesive crack, steady crack growth, and crack nucleation from a pressurized enclosure.

On optimization of heterogeneous materials for enhanced resistance to bulk fracture

We propose a novel approach to optimize the design of heterogeneous materials, with the goal of enhancing their effective fracture toughness under mode-I loading. The method employs a Gaussian processes-based Bayesian optimization framework to determine the optimal shapes and locations of stiff elliptical inclusions within a periodic microstructure in two dimensions. To model crack propagation, the phase-field fracture method with an efficient interior-point monolithic solver and adaptive mesh refinement, is used. To account for the high sensitivity of fracture properties to initial crack location with respect to heterogeneities, we consider multiple cases of initial crack and optimize the material for the worst-case scenario. We also impose a minimum clearance constraint between the inclusions to ensure design feasibility. Numerical experiments demonstrate that the method significantly improves the fracture toughness of the material compared to the homogeneous case.

Probabilistic investigation into brittle fracture of functionally graded materials using phase-field method

Modelling fracture using numerical techniques based on discrete modelling is complex as it requires tracking of progressing discontinuities in field parameters. Phase-field method for fracture is based on the variational framework and represents discontinuous crack surfaces by a damage variable that diffuses onto the crack surface. This approach offers the advantage of modelling crack where multiple crack nucleation, branching and coalescence can be determined without prior knowledge of the crack path. In this work, a probabilistic approach is used to predict the crack growth response in functionally graded material media when the mechanical properties and geometric parameters are random independent variables. Numerical implementation based on the standard phase-field method is employed in a finite element framework to model crack growth in functionally graded brittle materials. Peak failure loads are estimated within acceptable limits due to dispersion in system properties. Benchmark problems are solved to demonstrate the applicability of this technique. The proposed approach is advantageous as no further intensive computations are needed after the initial evaluation of probabilistic measures for predicting dispersion in fracture propagation when material and geometric properties exhibit scatter.

A phase-field model for mixed-mode elastoplastic fatigue crack

One of the most common damage modes of aero-engine hot-end components is crack. Because of the complicated geometry and harsh working environment, the crack is typically in a mixed-mode and under elastoplastic conditions. The phase-field fracture method is a numerical simulation method that has recently attracted extensive attention. However, attention toward material damage and crack initiation before the crack emerges is lacking to solve the problem of elastoplastic mixed-mode fatigue crack propagation. In this study, an enhanced phase-field fracture method addresses this issue and predicts the propagation behavior of the elastoplastic mixed-mode fatigue crack. The effect of some parameters of this model is analyzed. This achievement is a significant contribution to the safety of aircraft engines and a novel method that can be further improved to solve problems in the future.

Influence of Edge Crack Propagation on Critical Current Degradation of REBCO Tape Studied by Phase-Field Fracture Method and Current Shunting Model

For efficiency and cost savings during manufacturing, a fixed wide REBa <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Cu <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sub> -x (REBCO) tape is usually mechanically slit to a width required by varying applications, whereas the multiple edge cracks are induced in this process. In physics, it is widely recognized that the critical current of REBCO tape is degraded with tensile strain, and excessive strain causes crack growth. We are curious about how the edge crack grows under tensile strain, thereby further affecting the critical current performance. Therefore, we first introduced the phase-field method (PFM) to analyze the cracking and fracture behavior. After building a numerical model containing an edge crack and solving it by the finite element method, we presented the propagation path of the edge crack under tension and obtained the quantitative correlation between fracture ratio (crack depth/tape width) and tensile strain. Furthermore, since cracks impede the current flow and make the current shunted to the metal layer, resulting in critical current degradation, we adopted a current shunting model that considers the obstacle of cracks to investigate the relationship between critical current degradation and fracture ratio. Moreover, employing the intermediate variable of fracture ratio, we obtained the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${I}_c - \varepsilon $</tex-math></inline-formula> curve of REBCO tape. By comparing the calculated <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${I}_c - \varepsilon $</tex-math></inline-formula> curve with previous experimental data, the critical current degradation with tensile strain was very well interpreted by the propagation of the edge crack induced by mechanical slitting.

A “parallel universe” scheme for crack nucleation in the phase field approach to fracture

Crack nucleation is crucial in many industrial applications. The phase field method for fracture transforms the crack nucleation problem into a minimization problem of the sum of the elastic potential energy and the crack surface energy. Due to the polyconvexity of the formulation, starting from a crackless solid, a standard Newton iteration may lead to a solution with no crack, even though a cracked solution has a lower total energy. As such, the critical load for cracking is highly overestimated. Here, we propose an algorithm termed “parallel universe” algorithm to capture the global minimum. This algorithm has two key ingredients: (a) a necessary condition for cracking solely based on the current crackless solution, and (b) beginning from when this condition is met, Newton iteration with two initial guesses, a crackles one and a cracked one, will both be performed and the converged candidate solution with lower energy is accepted as the solution at that load step. Once the cracked candidate solution is accepted, the crackless one is discarded, i.e., only one universe is retained. This cracked initial guess is obtained only once for all load steps by solving a series of similar minimization problems with a progressively reduced critical crack energy release rate. Numerical examples with isotropic and anisotropic critical crack energy release rates indicate that the proposed algorithm is more reliable (as there is no need to retrace) and more efficient than the standard Newton iteration and a well-known backtracking algorithm.

A new hybrid phase-field model for modeling mixed-mode cracking process in anisotropic plastic rock-like materials

Mixed-mode cracking is widely encountered in rock-like geomaterials under compression-controlled loadings. Some pending issues still need to be addressed. This work focuses on a new hybrid phase-field fracture method for cracking behavior of anisotropic plastic materials. Several new approaches are introduced. Firstly, a unified decomposition method of elastic strain energy driving crack growth is proposed by combining the advantages of classical spectral and volumetric–deviatoric decomposition algorithms. Then a new damage field evolution criterion is established, allowing to considering a nonlinear interaction between two basic crack propagation modes. The contribution of plastic strain energy to cracking process is investigated by considering an anisotropic yield criterion and combined isotropic-kinematic hardening laws. The validation of the established phase-field model is assessed by a series of representative examples, showing different types of cracking patterns. Sensitivity analysis of some key parameters involved in the model is also carried out to clarify the respective impacts on the progressive cracking patterns. Finally, the effectiveness of the proposed approach is validated by comparison with experimental results.

Fracture Simulation of Ni–YSZ Anode Microstructures of Solid Oxide Fuel Cells Using Phase Field Method

The performance degradation of solid oxide fuel cells (SOFC) is directly related to the damage and fracture of electrode microstructures. In this study, the phase field fracture method is used to simulate the fracture of anode microstructures, and the effects of boundary constraints, thermal load, and Ni phase on the fracture of Ni–YSZ anode microstructures are investigated. Results show that tensile stresses occur in the Ni and YSZ phases whether above or below the reference temperature. The cracks propagate along the direction perpendicular to the first principal stress, showing a brittle fracture characteristic. When the microstructure is cooled, all cracks appear in YSZ phase, and almost all cracks initiate at the lowest point of YSZ–pore concave interface. When the microstructure is heated, the tensile first principal stress induces few cracks at local positions but will not make the cracks propagate continuously. The thermal mismatch between Ni and YSZ is not enough to induce cracks, and the fracture of electrode microstructure is more likely to be caused by external tensile load or the thermal mismatch between anode and electrolyte layers. The presence of Ni increases the stiffness of the microstructure, and solid phase’s disconnection reduces the strength of the microstructure.

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.

Phase field models of interface failure for bone application - evaluation of open-source implementations

Bone is a composite material where tubular osteon structures surrounded by thin interfaces reinforce the tissue at the microscale. These interfaces provide alternative weak paths for crack growth and give rise to potent extrinsic toughening mechanisms. Numerical models can help to gain a deeper understanding of fracture resistance in bone, however, capturing the interface mechanics is key. For this purpose, the phase field method for fracture is appealing due to its capability of capturing complex cracking phenomena. Still, it is far from well-established in the field of biomechanics. In this study, we evaluated a selection of recent open-source implementations of the phase field method to find the best approach for simulating crack growth in bone tissue. We also proposed a new method for correcting the interface fracture toughness in a phase field model. The selected implementations were compared using benchmark tests, including single edge notched tensile specimens with and without interfaces, as well as models representing typical bone geometries. We found the quasi-Newton monolithic solvers to be superior both in terms of computational speed and robustness compared to the evaluated staggered solvers. We also showed that correcting the fracture toughness of the interface is key for simulating crack patterns that are consistent with analytical predictions from linear elastic fracture mechanics.

Investigation on fracture behavior of polymer-bonded explosives under compression using a viscoelastic phase-field fracture method

Polymer-bonded explosives (PBXs) containing the energetic materials embedded in viscoelastic polymer matrix are often subjected to compressive loads in service, such as projectile penetration and drop. This study aims to investigate the fracture behavior of PBXs under compression by implementing a viscoelastic phase-field fracture (PFF) method. The mesh size and characteristic length scale of the present model are determined and further validated by comparing with experimental results. The influence of the viscoelasticity of the polymer matrix and heterogeneous microstructures on the fracture behavior of PBXs is comprehensively discussed. The results show that the strength and stiffness increase with an increasing strain rate; however, the failure strain increases at first and then decreases, which shows typical strain rate sensitivity. It is also found that the size and volume fraction of particulates have different effects on the fracture behavior of PBXs, i.e., the failure strength decreases with the increase of particulate size, but increases with the increase of particulate volume fraction. This study could lay the foundation for optimizing the structural design of high-performance PBXs.

Topology optimization for enhanced dynamic fracture resistance of structures

A topology optimization framework for improving the dynamic fracture resistance of structures is proposed. The phase field method for fracture is combined with Solid Isotropic Material with Penalization (SIMP) topology optimization. The topology optimization problem is defined as minimizing the fracture energy during the whole dynamic loading process, from initiation of cracks to full failure of the structure, under volume and compliance constraints. Semi analytical expressions of sensitivities in a dynamic context are provided to solve the topology optimization problem efficiently. Numerical examples involving structures subjected to impact loading are investigated. It is shown that the present framework allows a significant reduction of the fracture energy as compared to designs obtained by static optimization.

Experimentally validated phase-field fracture modeling of epoxy resins

In recent years the phase-field fracture (PFF) model has attracted much attention on failure problems in fiber-reinforced composite materials and structures. However, the discussion on the determination of input parameters remains unsettled. This work aims to seek a general solution for determining the length scale parameter lc and critical fracture energy density gf, the primary parameters of the PFF model. We design benchmark experiments, including uniaxial tensile and three-point bending tests, for two epoxy resins, PR 520 and LT-5078. The fracture structure responses of these two materials are investigated via the PFF method, emphasizing existing possible estimations for the length scale parameter lc. A calibration method is proposed for brittle fracture of epoxy resins, which combines benchmark experiments and numerical simulation trials. The input parameters of two epoxy resins are obtained using this calibration method. Simulated results are validated with experimental data qualitatively. Numerical predictions and experimental data are compared to show this method’s reliable estimation for the critical fracture energy density and length scale.

Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase-field fracture by nonlocal operator method

The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture method. The nonlocal governing equations are expressed as an integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate.