Phase Field Modeling Of Fracture Research Articles

Micromechanics-based variational phase-field modeling of fatigue fracture

In this paper, we extend the micromechanics-based phase-field model to simulate fatigue failure. The coupling of a micromechanics-based framework with the phase-field approach helps to differentiate between failure modes, by distinguishing between open and closed microcracks. This integrated framework links continuum field variables, such as plastic strain and damage variable, to micromechanical mechanisms like frictional sliding and microcrack opening. We first improve the algorithm’s stability during loading-unloading in the tensile regime through a modification of the plasticity evolution equations. Next, we incorporate fatigue damage accumulation and deterioration due to cyclic loading into the micromechanics-based phase-field model. A fatigue degradation function, driven by free energy accumulation, is introduced to degrade the fracture energy upon reaching a specified threshold during cyclic loading. Various cyclic loads are applied to benchmark tests, both with and without imperfections (e.g. holes, inclusions, voids), under plane strain conditions to capture diverse failure modes. The results demonstrate the model’s capability to accurately describe tensile, shear, and mixed-mode fracture under cyclic loading. Furthermore, the model effectively simulates key features of fatigue behavior, including crack nucleation, growth, and coalescence.

A micropolar phase-field model for size-dependent electro-mechanical fracture

This article proposes a micropolar phase-field model for size-dependent brittle fracture in solids under electro-mechanical loading conditions. Considering displacement, micro-rotation, electric potential, and phase-field variable as the kinematic descriptors and employing the virtual power principle, we derive a set of coupled governing partial differential equations (PDEs) for size-dependent solids. Invoking the first and second laws of thermodynamics, we determine the constitutive relations for the thermodynamic fluxes. Carrying out the finite element implementation of the derived governing PDEs using the open-source Gridap package in Julia, we demonstrate the efficacy of the proposed phase-field model through a few representative numerical examples. Especially the importance of the proposed model in incorporating the effect of relative rotation, i.e., the difference between macro- and micro-rotation, on the response of solids under electro-mechanical loading is shown that may not be possible with the existing non-local models such as strain-gradient or couple-stress approaches. To capture the experimentally observed size effects in solids under electro-mechanical loading, the proposed model does not demand higher-order continuity of finite element shape functions, unlike a typically used strain gradient model. To demonstrate the efficacy of the proposed model, we have compared our results against demanding experimental and numerical benchmark results available in the literature. We provide a parametric study to unravel the effect of different micropolar material parameters on the electro-mechanical response of a brittle solid. Interestingly, the proposed micropolar model is less sensitive to the phase-field length scale than the conventional non-polar phase-field models.

Microstructure-sensitive crystal plasticity and phase-field modeling of deformation and fracture in polycrystalline ice

The formation of crevasses in Greenland and Antarctica is primarily driven by the brittle failure of ice at low strain rates. Understanding this phenomenon requires exploring the effect of microstructural heterogeneity and anisotropic elastoplastic deformation on the fracture behavior of ice. Here, we have developed a microstructure-sensitive coupled crystal plasticity and phase-field model. This model allows us to study the plastic deformation, damage initiation and propagation under various strain rates in hexagonal open-packed polycrystalline ice, the prevailing phase on Earth. By employing a Bayesian optimization method, we derived the material parameters for the micromechanical model based on experimental results. The modeling results reveal that the activation of basal dislocations results in a brittle to ductile transition in ice at a low strain rate of 10-7s-1 under tension. At higher strain rates, the intrinsic plastic anisotropy of ice leads to heterogeneous deformation among grains and stress concentration near grain boundaries, triggering crack initiation and exacerbating the brittleness of ice. Under compression, cracks usually do not propagate throughout the specimen due to a significant decrease in stress around the crack tip upon propagation. Moreover, we found that the formation of the shear-induced basal texture at the bottom of ice layers and along glacier valley sides intensifies the brittleness of ice. This study provides insights into the micromechanical deformation and fracture mechanisms of ice, elucidating the intricate process of crevasse development in polar ice sheets.

Phase Field Coupled Finite Deformation Plasticity Formulation of Ductile Fracture With Nonlinear Kinematic Hardening and Modified Energy Release Function

ABSTRACTA ductile damage theory is presented by coupling the covariant formulation of finite deformation plasticity with the phase field modeling of fracture, including kinematic hardening for the ductile response of the materials. A phase field coupled nonlinear kinematic hardening equation is proposed in the reference configuration having equivalent representation through the Lie derivative of the kinematic hardening tensor by push‐forward operation in the spatial configuration, thus ensuring the satisfaction of frame invariance. To capture the correct physical response of the material by the phase field evolution equation, the fracture driving function, that is, the difference between the sum of elastic and plastic energies and threshold energy, after the damage initiation is modified by an energy release controlling function, which is an empirical relation of equivalent plastic strain. In defining the energy release controlling function, well‐defined points with physical significance in the experimental load versus displacement curve are used. To simulate the response of the material in relatively large time steps, a modified staggered scheme is presented, evaluating the fracture driving and energy release controlling functions from the previous converged step and using the updated phase field variable in the weak form of the momentum balance equation. To quantify different material parameters from available experimental results in the literature, the developed phase field coupled elasto‐plastic model uses a neural network optimization procedure consisting of neural network training together with optimization in MATLAB and finite element model evaluation in Abaqus user element subroutine UEL. Model capabilities are demonstrated by simulating the crack propagation in complex 3D geometries such as the second and third Sandia Fracture Challenges.

Orientation-dependent deformation and failure of micropillar shape memory ceramics: A 3D phase-field study

Some microscopic samples of zirconia-based shape memory ceramics (SMCs) have shown full martensitic phase transformation (MPT) over multiple loading cycles without cracking. However, the occurrence of MPT is strongly influenced by grain orientation. Depending on the specific grain orientation relative to the loading direction, alternative mechanisms such as plastic slip and fracture may emerge. This study introduces a phase-field (PF) based framework that integrates a PF-MPT model, a PF fracture model, and a crystal viscoplasticity model to investigate the effects of grain orientation on MPT, plastic slip, and fracture mechanisms in SMC micropillars. Single crystal micropillars are created to distinguish the orientations that facilitate each mechanism. A wide range of grain orientations are found to predominantly exhibit MPT. Micropillars with grain orientations close to the (100) and (001) directions primarily experience fracture, with minimal plastic slip. Additionally, samples oriented along the (110) direction show a significant amount of plastic slip. A pole figure is constructed to elucidate the interplay between MPT, cracking, and plastic slip under compressive loading conditions. This research provides valuable insights into the intricate behavior of SMCs under different loading scenarios, crucial for optimizing their performance in practical applications.

Phase-field approach for precise fracture tracking in anisotropic rocks: Integrating orthotropy-based energy decomposition and two-fold symmetric fracture toughness

Rock formations are known to exhibit material anisotropy, both in terms of elastic and fracture properties. This means that the fracture path in such formations is not a priori known but rather a complex unknown that requires robust numerical techniques to predict accurately. In this context, the phase-field model is considered particularly effective, provided that certain physical considerations are carefully adjusted to align with the physics of the problem. While addressing elastic anisotropy is well-established, the tension–compression asymmetry necessary to inhibit crack interpenetration in phase-field fracture models needs to account for the specific material anisotropy. Additionally, to accurately capture crack propagation, it is critical to simultaneously account for orientation-dependent fracture toughness in such materials. To address this, the present study employs an anisotropic phase-field model that integrates the generalized spectral decomposition proposed in the literature for orthotropic materials with a two-fold symmetric fracture toughness, to predict the fracture trajectories in rock-type samples under fixed mixed-mode loading ratios. While each of the two aspects has primarily been applied to model orthotropic plates under simple tensile and shearing loading conditions in the literature, here we study their applicability in complex loading scenarios. To this end, the experimental data from notched semi-circular specimens of Grimsel Granite undergoing complex mixed-mode loading obtained in our previous work is considered. We focus on two given mode-mixity ratios and perform numerical studies. Our results emphasize the importance of considering this generalized decomposition for phase-field modeling of fracturing in rock-type materials, particularly under loading conditions where the crack might otherwise be unrealistically driven into the compressive region. Although certain features are well captured by considering anisotropy in elasticity alone, our findings demonstrate that incorporating a two-fold symmetric fracture toughness proves to be advantageous for more precise tracking of the fracture path.

Cracking Mechanism and Inhibition Strategies of Polycrystalline NCM Electrode Particles.

Developing a high-energy-density cathode material (LiNi1-x-yCoxMnyO2, NCM) for lithium-ion batteries is crucial to the electric vehicle and energy storage industries. However, the continuous insertion/extraction of Li+ generates diffusion-induced stress, causing NCM particles to crack or even pulverize, leading to battery capacity loss and limiting its wider commercial application. Current experimental studies are primarily postmortem examinations, and it is difficult to capture the particle cracking evolution. Simulation studies frequently ignore or simplify anisotropic volume contraction, demonstrating an insufficient understanding of the cracking mechanism of NCM polycrystalline particles, and cracking prevention strategies still need improvement. Therefore, we develop an anisotropic polycrystalline fracture phase-field model (AP-FPFM) that focuses on the anisotropic volume contraction of primary particles and precisely generates grain boundary distribution, coupling with Li+ diffusion, mechanical stress, and particle cracking. We employ AP-FPFM to demonstrate the behavior and mechanism of NCM polycrystalline particle cracking and illustrate the necessity and importance of anisotropic volume contraction to understand particle cracking. Furthermore, we explore the effects of average primary particle size, secondary particle size, and core-shell structure modulation on crack initiation and propagation and propose strategies to inhibit or migrate NCM polycrystalline particle cracking. This work provides theoretical support for revealing the cracking mechanism of anisotropic polycrystalline NCM particles and supplying optimization strategies to suppress particle cracking and improve the mechanical stability.

Fatigue crack growth in functionally graded materials using an adaptive phase field method with cycle jump scheme

Functionally graded materials provide versatility in adjusting the volume fractions of constituent materials to meet specific design requirements. However, this customization often introduces mode-mixity at the crack tip, posing challenges in predicting fracture under cyclic loading with discrete approaches and computationally expensive with conventional phase-field fracture models. To address these issues, this paper introduces an adaptive phase-field fracture formulation with cycle jump scheme to elegantly predict fatigue crack nucleation and growth in functionally graded materials. Within this framework, the effective properties at a point are estimated using the Mori–Tanaka homogenization scheme, while the crack growth due to cyclic load is captured by incorporating an additional fatigue degradation parameter. Moreover, the computational efficiency of the proposed framework is improved through an adaptive mesh refinement and explicit cycle jump scheme. The adaptive refinement scheme utilizes an error indicator derived from both the displacement solution and phase-field variable. The adaptive refinement scheme is integrated with efficient quadtree decomposition, which generates a hierarchical mesh structure. Hanging nodes resulting from the quadtree decomposition are efficiently handled using a polygonal finite element method. The proposed framework is validated against experimental and numerical results reported in the literature. Furthermore, we investigate the fatigue crack growth resistance across a broad range of material gradation directions, gaining valuable insights and identifying functionally graded materials with high fatigue resistance.

Fracture as an Emergent Phenomenon

The mechanics of fracture propagation provides essential knowledge for the risk tolerance design of devices, structures, and vehicles. Techniques of free energy minimization provide guidance, but have limited applicability to material systems evolving away from equilibrium. Experimental evidence shows that the material response depends on driving forces arising from mechanical fields. Recent years have witnessed the development of new methods for modeling complex dynamic and quasistatic fracture. New approaches may differ remarkably from previous ones, as they involve implicit coupling between damaged and undamaged states, allowing fracture to be modeled as emergent phenomena.The focus of this workshop is on the most advanced techniques for modeling fracture, represented by eigenerosion methods, variational approaches, phase field fracture models, and non-local approaches. Technical progress is contingent on the further development of the mathematical framework underlying these techniques. This is necessary for accurate and reliable computational modeling of fracture for multiple freely propagating cracks. The objective of this workshop is to mathematically identify and discuss open issues related to fracture modeling and to highlight recent advances. Addressing fundamental issues will foster exchange between the different communities, essential for advancing the field.

Effect of nitrogen-doped type on fracture toughness improvement and crack growth resistance of carbon nanotube/epoxy nanocomposites: Combined multiscale analysis approach

Recently, nitrogen-doped carbon nanotubes (N-doped CNTs) have received great attention in nanocomposite design. It has become highly necessary to develop predictive models to elucidate their toughning behavior. In this study, the effects of CNTs with three different types of N-doped functional groups (quaternary, pyrrolic, and pyridinic) on the fracture toughness (FT) and crack growth of polymer nanocomposites are predicted using a multiscale analysis approach. To scale up from the nanoscale to the macroscale, a multiscale analysis approach integrating molecular dynamics, micromechanics theory, linear fracture mechanics theory, and a phase-field fracture model (PFFM) is adopted. The toughness enhancement trends of the three different types of N-doped functional groups were quantified by considering four toughening mechanisms (CNT debonding, plastic nanovoid growth, CNT pull-out, and CNT rupture), and compared with experimental result. The results show that the excellent interphase and interfacial properties of quaternary and pyridinic functional groups significantly improve the FT and crack growth resistance of N-doped CNT/epoxy nanocomposites. Our study provides high-performance solutions for experimental studies pertaining to the FT and crack growth of N-doped CNT/epoxy nanocomposites.

Efficient BFGS quasi-Newton method for large deformation phase-field modeling of fracture in hyperelastic materials

The prediction of crack propagation in materials is a crucial problem in solid mechanics, with many practical applications ranging from structural integrity assessment to the design of advanced materials. The phase-field method has emerged as a powerful tool for modeling crack propagation in materials, due to its ability to accurately capture the propagation of cracks. However, current phase-field algorithms suffer from the elevated computational cost associated with the so-called staggered solution scheme, which requires extremely small time increments to advance the crack due to its inherent conditional stability. In this paper, we present, for the first time, a quantitative analysis detailing the numerical implementation and comparison of two common solution strategies for the coupled large-deformation solid-mechanics-phase-field problem, namely the quasi-Newton based Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) and the full Newton based alternating minimization (or staggered) (AM/staggered) algorithm. We demonstrate that the BFGS algorithm is a more efficient and advantageous alternative to the traditional AM/staggered approach for solving coupled large-deformation solid-mechanics-phase-field problems. Our results highlight the potential of the quasi-Newton BFGS algorithm to significantly reduce the computational cost of predicting crack propagation in hyperelastic materials while maintaining the accuracy and robustness of the phase-field method.

Formation mechanism of radial and circular cracks promoted by delamination in drying silica colloidal deposits.

Cracks with radial and circular patterns are appealing in nature and industry. Although morphologies and propagation conditions of cracks are extensively studied, the formation mechanism of crack pattern by the interaction of channel fracture and interfacial delamination remains elusive. Here, we present the transition of radial to coexisting radial and circular crack patterns when the thickness of colloidal deposits on both hard and soft substrates exceeds a critical value, through the colloidal volume fraction dependence. In addition, a thickness-dependent phase diagram from radial crack to coexistence of radial and circular cracks was constructed with respect to the radius and the volume fractions of silica colloidal deposits. A phase-field fracture model is developed to elucidate how the formation of radial cracks is facilitated by simultaneous delamination. The warping-induced radial tensile stress at the bottom surface of the striped deposit is proportional to the thickness. It leads to subsequent nucleation and growth of circular cracks in thick deposits. This work provides insight into the formation mechanism of complex crack patterns in drying colloidal deposits and revolutionizes the design space of crack-based micro-nano structures.

A phase-field gradient-based energy split for the modeling of brittle fracture under load reversal

In the phase-field modeling of fracture, the search for a physically reasonable and computationally feasible criterion to split the elastic energy density into fractions that may or may not contribute to crack propagation has been the subject of many recent studies. Within this context, we propose an energy split – or energy decomposition – aimed at accurately representing the evolution of a crack under load reversal. To this purpose, two key assumptions are made. First, the damage gradient direction is interpreted as being representative of the normal-to-crack direction, as already assumed in previous works in the literature. The second assumption consists of considering the sign of the projection of the stress tensor onto the damage gradient direction at a point as an indicator of whether this point should behave as an opening or as a closing crack. We associate the latter case (crack closing) to both (a) a complete recovery of elastic energy density of the intact material (i.e., perfectly rough crack surfaces) and (b) a zero crack driving force at that point. The first case (crack opening) is treated classically as a damageable material point at which damage can increase. The implementation of the proposed approach turns out to be remarkably simple and computationally robust. For the evaluation of the displacements and damage gradients at nodes, the classical Z2 technique is used, and a new effective and computationally convenient iterative strategy is implemented to guarantee convergence of the staggered scheme. Four examples are presented in order to assess the suitability of the present model by using both AT1 and AT2 regularization models. Results show the desired effect of limiting crack propagation to prevailing tensile states, as well as of recovering the initial intact stiffness upon load reversal, even when two of the most common energy splits fail.

A phase-field length scale insensitive mode-dependent fracture model for brittle failure

This article presents a consistent phase-field length scale insensitive mode-dependent fracture model for brittle failure by proposing mode-factor dependent degradation functions that incorporate the effect of two additional fracture parameters, namely the mode-II critical energy release rate and mode-II fracture strength, on the overall mechanical response. Using the proposed mode-factor dependent degradation functions and employing a recently proposed modified strain decomposition scheme, we provide analytical expressions for the mode-I and mode-II fracture strengths corresponding to the mode-dependent parts of the elastic energy. Adopting a modified Benzeggagh–Kenane (B–K) criteria, we propose mode-dependent driving forces by deriving expressions for the critical energy release rate corresponding to individual fracture modes. The proposed model provides a consistent coupling between the different fracture modes and can thus predict fracture for all possible mode-mixity ratios. A parametric study is carried out to unravel the effect of mixed-mode fracture parameters on the mechanical response of isotropic materials by considering a few representative numerical examples. The numerical results from the proposed model show an excellent agreement with experimental results reported in the literature.

Mechanical Failure of Core-Shell Cathode Particles: The Effects of Concentration-Dependent Material Properties and Phase Field Fracture Modelling

The use of core-shell cathode particles in lithium-ion batteries is an attractive approach to enhancing energy density whilst retaining lifetime, through reducing undesired reactions at the electrode-electrolyte interface and limiting the volume change of the electrode particles. However, mechanical failure through the fracture and debonding of the core-shell interface is a major challenge. In this work, we employ a coupled finite-element model to predict and mitigate the mechanical failure of core-shell cathode structures, taking as example a particle of NMC811 (core) coated with NMC111 (shell). In particular, we focus on two aspects:The first one involves the assumptions of material properties as inputs of the model. The material properties are often considered constant by battery modelling researchers, yet these parameters can vary significantly during charge/discharge. For example, experiments have unveiled a three-orders-of-magnitude drop in the diffusion coefficient of NMC materials during discharge [1]. Here, we incorporate material properties obtained from experimental data, including concentration-dependent diffusion coefficient obtained from GITT measurement and partial molar volume derived from in situ X-ray diffraction data. Our results indicate that when assuming a concentration-dependent partial molar volume, the maximum values of tensile hoop stress in the shell are nearly three times lower than those predicted with constant average properties, diminishing the likelihood of fracture.When accounting for concentration-dependent diffusion coefficient, large concentration gradient is observed near the outer surface of the core due to reduced lithium mobility at high states of lithiation, hindering full electrode capacity utilisation. The significant concentration gradient and capacity underutilisation align with direct observations from experiments [2]. Notably, these phenomena cannot be captured in modelling if assuming constant diffusion coefficient. These findings offer new perspectives on the performance and design of core-shell electrode particles. Safe design maps in terms of core size and shell thickness to mitigate fracture and debonding are obtained.The second focus of this work is the utilisation of phase field fracture model in predicting cracking behaviours within the core-shell particle. Phase field modelling provides a continuous representation of cracks, enabling the tracking of complex cracking patterns [3]. We predict cracking behaviors caused by volume changes during charge/discharge, as well as inaccessible capacity and newly created active surface area caused by cracking. Our multiphyics model can hopefully act as a catalyst, accelerating the development process of the core-shell structured cathode particles and shortening the time to market of advanced batteries with extended lifetime.

A computational approach for phase-field model of quasi-brittle fracture under dynamic loading

AbstractA computational model is formulated for studying dynamic crack propagation in quasi-brittle materials exposed to time-dependent loading conditions. Under such conditions, inertial effects of structural components play an important role in modelling crack propagation problems. The computational model is proposed within the theory of regularised cracks which uses a damage-like internal variable. Here, fracture considers phase-field damage which gives rise to a material degradation in a narrow material strip defining the regularised crack. Based on the energy formulation using the Lagrangian of the system, the proposed computational approach introduces a staggered scheme adopted to solve the coupled system and providing it in a variational form within the time stepping procedure. The numerical data are obtained by quadratic programming algorithms implemented together with a finite element code.

Verification of strain energy splits of phase field fracture model using Westergaard’s problem under mixed-mode loading

One challenge in verifying strain energy splits in phase field fracture models is the lack of reference problems with exact solutions in mixed-mode loading that could serve as benchmark problems. In this work, different strain energy splits used in the variational phase field model have been applied to the classical Westergaard’s problem. In contrast to other benchmark problems used in the literature, the proposed problem not only includes the singular term as a boundary condition but also implicitly considers all terms. To the authors’ knowledge, this is the first time in the literature that this reference problem has been used to rigorously verify the accuracy of different energy splits of the phase field model. Several critical factors such as the pre-crack definition, mesh size, mixed mode ratio and the length scale parameter have been analysed. A good correlation between the numerical estimations and the analytical predictions was found for some pre-crack definition approaches under mode I. However, we demonstrated that the compatible regularisation length scale and mesh size are significantly lower under mode II loading (i.e., when the crack kinks) than under mode I.

Anisotropic damage evolution in solid fractures: A novel phase field approach with multiple failure criteria and directional-dependent structural tensor

This study proposes a novel phase-field fracture model based on unified phase field theory, aiming to overcome current limitations in simulating material complex fracture behaviors. Through this model, analytical solutions for two-dimensional bars subjected to tensile or compressive stresses are provided, enabling the coupling of multiple failure criteria and further proficient simulation of mode-I, mode-II, and mixed mode-I/II fractures, effectively addressing challenges faced in modelling materials with different or complex failure modes under various loading conditions. Furthermore, to account for the strong anisotropic failure behavior of materials, a novel directional-dependent structural tensor is proposed. The tensor correlates fracture energy with crack surface orientation, facilitating precise characterization of material damage evolution with multiple potential crack orientations. This tensor ensures the consistency of phase-field fracture evolution with predefined fracture patterns. The effectiveness of the proposed model is validated through case studies, emphasizing its robustness and superior predictive capability in capturing fracture behavior under various conditions. This research provides a more accurate and universally applicable approach for simulating material failure, particularly for complex or multiple failure mode material failure simulations.

Phase field modelling of elastic-plastic fatigue fracture of oil and gas pipeline

This paper established a fatigue fracture phase-field model (PFM) to evaluate fatigue damage evolution and crack propagation in oil and gas pipeline. To address inaccuracies in damage evolution, a threshold of the elastic-plastic fracture energy was introduced in the proposed PFM. Using the finite element method, the PFM was applied to simulate fatigue crack growth. Results from compact tension (CT) specimen of the X56 gas pipeline steel demonstrated that the da/dN-ΔK curve from the current PFM, accounting for plasticity, aligned more closely with experimental results than the elastic PFM. The fatigue crack propagation and fatigue life of the X80 gas pipeline with different defects of the same depth were also analyzed. The results indicated that triangular defects significantly impacted the fatigue life of the X80 gas pipeline. Finally, a model of X60 pipeline with various initial defects was developed to validate the effectiveness of the proposed PFM for full-scale pipeline fatigue fracture by comparing it to experimental a-N curves. The simulation results indicated that the distance and angle between two initial defects in the pipeline significantly influenced the propagation of fatigue cracks and the pipeline’s service life. These findings of this paper can serve as a reference for estimating the service life of gas and oil pipelines.

Calibration and validation of a phase-field model of brittle fracture within the damage mechanics challenge

In the context of the Damage Mechanics Challenge, we adopt a phase-field model of brittle fracture to blindly predict the behavior up to failure of a notched three-point-bending specimen loaded under mixed-mode conditions. The beam is additively manufactured using a geo-architected gypsum based on the combination of bassanite and a water-based binder. The calibration of the material parameters involved in the model is based on a set of available independent experimental tests and on a two-stage procedure. In the first stage an estimate of most of the elastic parameters is obtained, whereas the remaining parameters are optimized in the second stage to minimize the discrepancy between the numerical predictions and a set of experimental results on notched three-point-bending beams. The good agreement between numerical predictions and experimental results in terms of load–displacement curves and crack paths demonstrates the predictive ability of the model and the reliability of the calibration procedure.