Academic literature on the topic 'Phase field modeling of brittle fracture'

Une bonne tenue mécanique des structures du génie civil en béton armé sous chargements dynamiques sévères est primordiale pour la sécurité et nécessite une évaluation précise de leur comportement en présence de propagation dynamique de fissures. Dans ce travail, on se focalise sur la modélisation constitutive du béton assimilé à un matériau élastique-fragile endommageable. La localisation des déformations sera régie par un modèle d'endommagement à gradient où un champ scalaire réalise une description régularisée des phénomènes de rupture dynamique. La contribution de cette étude est à la fois théorique et numérique. On propose une formulation variationnelle des modèles d'endommagement à gradient en dynamique. Une définition rigoureuse de plusieurs taux de restitution d'énergie dans le modèle d'endommagement est donnée et on démontre que la propagation dynamique de fissures est régie par un critère de Griffith généralisé. On décrit ensuite une implémentation numérique efficace basée sur une discrétisation par éléments finis standards en espace et la méthode de Newmark en temps dans un cadre de calcul parallèle. Les résultats de simulation de plusieurs problèmes modèles sont discutés d'un point de vue numérique et physique. Les lois constitutives d'endommagement et les formulations d'asymétrie en traction et compression sont comparées par rapport à leur aptitude à modéliser la rupture fragile. Les propriétés spécifiques du modèle d'endommagement à gradient en dynamique sont analysées pour différentes phases de l'évolution de fissures : nucléation, initiation, propagation, arrêt, branchement et bifurcation. Des comparaisons avec les résultats expérimentaux sont aussi réalisées afin de valider le modèle et proposer des axes d'amélioration<br>In civil engineering, mechanical integrity of the reinforced concrete structures under severe transient dynamic loading conditions is of paramount importance for safety and calls for an accurate assessment of structural behaviors in presence of dynamic crack propagation. In this work, we focus on the constitutive modeling of concrete regarded as an elastic-damage brittle material. The strain localization evolution is governed by a gradient-damage approach where a scalar field achieves a smeared description of dynamic fracture phenomena. The contribution of the present work is both theoretical and numerical. We propose a variationally consistent formulation of dynamic gradient damage models. A formal definition of several energy release rate concepts in the gradient damage model is given and we show that the dynamic crack tip equation of motion is governed by a generalized Griffith criterion. We then give an efficient numerical implementation of the model based on a standard finite-element spatial discretization and the Newmark time-stepping methods in a parallel computing framework. Simulation results of several problems are discussed both from a computational and physical point of view. Different damage constitutive laws and tension-compression asymmetry formulations are compared with respect to their aptitude to approximate brittle fracture. Specific properties of the dynamic gradient damage model are investigated for different phases of the crack evolution: nucleation, initiation, propagation, arrest, kinking and branching. Comparisons with experimental results are also performed in order to validate the model and indicate its further improvement

La fabrication additive attire une attention croissante en raison de ses avantages en termes de flexibilité de modélisation et de facilité de conception de microstructures complexes. Nous avons constaté qu'en manipulant la stratégie d'impression, les échantillons imprimés par dépôt de fusion de polycarbonate peuvent présenter un comportement fortement anisotrope en termes de résistance à la rupture, tout en conservant des propriétés isotropes en termes d'élasticité.Le focus de cette thèse est d'explorer le comportement en matière de rupture dans des milieux élastiques isotropes présentant une ténacité de rupture anisotrope, en utilisant une combinaison d'investigations expérimentales et de simulations numériques. Dans la partie expérimentale, nous examinons la propagation des fissures dans diverses conditions de chargement en utilisant des géométries d'échantillons variées, englobant à la fois le Mode I et le Mode I+II. Dans la partie numérique, nous adoptons la modélisation de la fissuration fragile par champ de phase basée sur l'approche variationnelle, en utilisant des données expérimentales pour l'étalonnage et l'identification des paramètres numériques. À travers ces méthodologies complètes, notre objectif est de favoriser une compréhension plus profonde de l'interaction entre les motifs d'impression et la sélection des trajectoires de fissures. Cette compréhension a des implications significatives pour guider et gérer la propagation des fissures dans les composants fabriqués par fabrication additive. De plus, nous adoptons les critères classiques basés sur le taux de restitution d'énergie maximale généralisé pour améliorer notre compréhension de la sélection des trajectoires de fissures et de la force critique correspondante.Dans la dernière partie de cette thèse, nous présentons quelques investigations préliminaires concernant l'éventuelle émergence d'un motif de fissure en Zig-Zag dans des spécimens imprimés en 3D. De plus, nous plongeons en profondeur dans le comportement de rupture des spécimens imprimés sous chargement cyclique, offrant une comparaison exhaustive entre les observations expérimentales et les prévisions numériques<br>Additive manufacturing is receiving increasing attention due to its advantages in terms of modelling flexibility and allowing to easily design complex micro-structures. Through the manipulation of the printing strategy, we observed that fused deposition of polycarbonate can result in printed samples showcasing a distinct anisotropic behavior in fracture toughness, all the while retaining isotropic properties in elasticity.This thesis is dedicated to investigating fracture behavior within isotropic elastic media with anisotropic fracture toughness. The approach involves a combination of fracture experiments and numerical simulations. In the experimental part, we examine crack propagation under various loading conditions using diverse sample geometries, encompassing both Mode I and Mode I+II loading condition. In the numerical part, we adopt the phase-field modeling of brittle fracture based on a variational approach, using experimental data for calibrating and identification of the numerical parameters. Through these comprehensive methodologies, our objective is to foster a deeper comprehension of the interplay between printing patterns and the selection of crack paths. This understanding holds significant implications for guiding and controlling crack propagation in additive manufacturing-produced components. Besides, we adopted the classical based criteria Generalized Maximum Energy Release Rate to enhance our understanding of crack path selection and the relevant critical force.In the last part of this thesis, we presents some preliminary investigations regarding the potential emergence of Zig-Zag crack patterns in 3D printed specimens. Additionally, we delve extensively into the fracture behavior of printed specimens under cyclic loading, offering a comprehensive comparison between experimental observations and numerical forecasts

The objective of this paper is to propose a 2-D time-independent phase-field model with validating its performance as well as applying it for simulating existing representative experiments. Firstly, the section of the literature review provides an overview of quasi-brittle material and brittle fracture behaviours, as well as the existing FE models from both discontinuous and continuous approaches for simulating fracture behaviours. Next, the governing equations of the proposed phase-field model are determined, which are based on traditional Griffith’s theory as well as a specific variational method evolved from that. The proposed model is implemented in Abaqus. In particular, the implementation is achieved by using the User Subroutine in order to take the phase-field into account. The proposed model is validated by simulating a pure-tension and a pure shear test. In this part, not only the effect of discretisation but also the effects of length parameter and energy release rate has been discussed, of which the latter effect is exclusive in phase-field method. Finally, the validated model is used for simulating two sets of existing experiments, including a mixed-mode test and a series of Brazilian disks test. The results in both validation and simulation part indicate that the proposed model can successfully simulate both crack initiation and propagation in these cases, and good qualitative agreement with theoretical or experimental results can be observed.

Les simulations numériques des fissures fragiles par les modèles d’endommagement à gradient deviennent main- tenant très répandues. Les résultats théoriques et numériques montrent que dans le cadre de l’existence d’une pre-fissure la propagation suit le critère de Griffith. Alors que pour le problème à une dimension la nucléation de la fissure se fait à la contrainte critique, cette dernière propriété dimensionne le paramètre de longueur interne.Dans ce travail, on s’attarde sur le phénomène de nucléation de fissures pour les géométries communément rencontrées et qui ne présentent pas de solutions analytiques. On montre que pour une entaille en U- et V- l’initiation de la fissure varie continument entre la solution prédite par la contrainte critique et celle par la ténacité du matériau. Une série de vérifications et de validations sur diffèrent matériaux est réalisée pour les deux géométries considérées. On s’intéresse ensuite à un défaut elliptique dans un domaine infini ou très élancé pour illustrer la capacité du modèle à prendre en compte les effets d’échelles des matériaux et des structures.Dans un deuxième temps, ce modèle est étendu à la fracturation hydraulique. Une première phase de vérification du modèle est effectuée en stimulant une pré-fissure seule par l’injection d’une quantité donnée de fluide. Ensuite on étudie la simulation d’un réseau parallèle de fissures. Les résultats obtenus montrent qu’il a qu’une seule fissure qui se propage et que ce type de configuration minimise mieux l’énergie la propagation d’un réseau de fractures. Le dernier exemple se concentre sur la stabilité des fissures dans le cadre d’une expérience d’éclatement à pression imposée pour l’industrie pétrolière. Cette expérience d’éclatement de la roche est réalisée en laboratoire afin de simuler les conditions de confinement retrouvées lors des forages.La dernière partie de ce travail se concentre sur la rupture ductile en couplant le modèle à champ de phase avec les modèles de plasticité parfaite. Grâce à l’approche variationnelle du problème on décrit l’implantation numérique retenue pour le calcul parallèle. Les simulations réalisées montrent que pour une géométrie légèrement entaillée la phénoménologie des fissures ductiles comme par exemple la nucléation et la propagation sont en concordances avec ceux reportées dans la littérature<br>Phase-field models, sometimes referred to as gradient damage, are widely used methods for the numerical simulation of crack propagation in brittle materials. Theoretical results and numerical evidences show that they can predict the propagation of a pre-existing crack according to Griffith’s criterion. For a one- dimensional problem, it has been shown that they can predict nucleation upon a critical stress, provided that the regularization parameter is identified with the material’s internal characteristic length.In this work, we draw on numerical simulations to study crack nucleation in commonly encountered geometries for which closed-form solutions are not available. We use U- and V-notches to show that the nucleation load varies smoothly from the one predicted by a strength criterion to the one of a toughness criterion when the strength of the stress concentration or singularity varies. We present validation and verification of numerical simulations for both types of geometries. We consider the problem of an elliptic cavity in an infinite or elongated domain to show that variational phase field models properly account for structural and material size effects.In a second movement, this model is extended to hydraulic fracturing. We present a validation of the model by simulating a single fracture in a large domain subject to a control amount of fluid. Then we study an infinite network of pressurized parallel cracks. Results show that the stimulation of a single fracture is the best energy minimizer compared to multi-fracking case. The last example focuses on fracturing stability regimes using linear elastic fracture mechanics for pressure driven fractures in an experimental geometry used in petroleum industry which replicates a situation encountered downhole with a borehole called burst experiment.The last part of this work focuses on ductile fracture by coupling phase-field models with perfect plasticity. Based on the variational structure of the problem we give a numerical implementation of the coupled model for parallel computing. Simulation results of a mild notch specimens are in agreement with the phenomenology of ductile fracture such that nucleation and propagation commonly reported in the literature

The unique electro-mechanical coupling properties of ferroelectrics make them ideal materials for use in micro-devices as sensors, actuators and transducers. Nevertheless, because of the intrinsic brittleness of ferroelectrics, the optimal design of the electro-mechanical devices is strongly dependent on the understanding of the fracture behavior in these materials. Fracture processes in ferroelectrics are notoriously complex, mostly due to the interactions between the crack tip stress and electric fields and the localized switching phenomena in this zone (formation and evolution of domains of different crystallographic variants). Phase-field models are particularly interesting for such a complex problem, since a single partial differential equation governing the phase-field accomplishes at once (1) the tracking of the interfaces in a smeared way (cracks, domain walls) and (2) the modeling of the interfacial phenomena such as domain-wall energies or crack face boundary conditions. Such a model has no difficulty for instance in describing the nucleation of domains and cracks or the branching and merging of cracks. Furthermore, the variational nature of these models makes the coupling of multiple physics (electrical and mechanical fields in this case) very natural. The main contribution of this thesis is to propose a phase-field model for the coupled simulation of the microstructure formation and evolution, and the nucleation and propagation of cracks in single crystal ferroelectric materials. The model naturally couples two existing energetic phase-field approaches for brittle fracture and ferroelectric domain formation and evolution. The finite element implementation of the theory is described. Simulations show the interactions between the microstructure and the crack under mechanical and electro-mechanical loadings. Another objective of this thesis is to encode different crack face boundary conditions into the phase-field framework since these conditions strongly affect the fracture behavior of ferroelectrics. The smeared imposition of these conditions are discussed and the results are compared with that of sharp crack models to validate the proposed approaches. Simulations show the effects of different conditions, electro-mechanical loadings and media filling the crack gap on the crack propagation and the microstructure of the material. In a third step, the coupled model is modified by introducing a crack non-interpenetration condition in the variational approach to fracture accounting for the asymmetric behavior in tension and compression. The modified model makes it possible to explain anisotropic crack growth in ferroelectrics under Vickers indentation loading. This model is also employed for the fracture analysis of multilayer ferroelectric actuators, which shows the potential of the model for future application. The coupled phase-field model is also extended to polycrystals by introducing realistic polycrystalline microstructures in the model. Inter- and trans-granular crack propagation modes are observed in the simulations. Finally and for completeness, the phase-field theory is extended for the simulation of conducting cracks and some preliminary simulations are also performed in three dimensions. Salient features of the crack propagation phenomenon predicted by the simulations of this thesis are directly compared with experimental observations.<br>Los materiales ferroeléctricos poseen únicas propiedades electro-mecánicas y por eso se utilizan para los micro-dispositivos como sensores, actuadores y transductores. No obstante, debido a la fragilidad intrínseca de los ferroeléctricos, el diseño óptimo de los dispositivos electro-mecánicos es altamente dependiente de la comprensión del comportamiento de fractura en estos materiales. Los procesos de fractura en ferroeléctricos son notoriamente complejos, sobre todo debido a las interacciones entre campos de tensión y eléctricos y los fenómenos localizados en zona de fractura (formación y evolución de los dominios de las diferentes variantes cristalográficas). Los modelos de campo de fase son particularmente útiles para un problema tan complejo, ya que una sola ecuación diferencial parcial que gobierna el campo de fase lleva a cabo a la vez (1) el seguimiento de las interfaces de una manera suave (grietas, paredes de dominio) y (2) la modelización de los fenómenos interfaciales como las energías de la pared de dominio o las condiciones de las caras de grieta. Tal modelo no tiene ninguna dificultad, por ejemplo en la descripción de la nucleación de los dominios y las grietas o la ramificación y la fusión de las grietas. Además, la naturaleza variacional de estos modelos facilita el acoplamiento de múltiples físicas (campos eléctricos y mecánicos en este caso). La principal aportación de esta tesis es la propuesta de un modelo campo de fase para la simulación de la formación y evolución de la microestructura y la nucleación y propagación de grietas en materiales ferroeléctricos. El modelo aúna dos modelos de campo de fase para la fractura frágil y para la formación de dominios ferroeléctricos. La aplicación de elementos finitos a la teoría es descrita. Las simulaciones muestran las interacciones entre la microestructura y la fractura del bajo cargas mecánicas y electro-mecánicas. Otro de los objetivos de esta tesis es la codificación de diferentes condiciones de contorno de grieta porque estas condiciones afectan en gran medida el comportamiento de la fractura de ferroeléctricos. La imposición de estas condiciones se discuten y se comparan con los resultados de modelos clasicos para validar los modelos propuestos. Las simulaciones muestran los efectos de diferentes condiciones, cargas electro-mecánicas y medios que llena el hueco de la grieta en la propagación de las fisuras y la microestructura del material. En un tercer paso, el modelo se modifica mediante la introducción de una condición que representa el comportamiento asimétrico en tensión y compresión. El modelo modificado hace posible explicar el crecimiento de la grieta anisotrópica en ferroeléctricos. Este modelo también se utiliza para el análisis de la fractura de los actuadores ferroeléctricos, lo que demuestra el potencial del modelo para su futura aplicación. El modelo se extiende también a policristales mediante la introducción de microestructuras policristalinas realistas en el modelo. Modos de fractura inter y trans-granulares de propagación se observan en las simulaciones. Por último y para completar, la teoría del campo de fase se extiende para la simulación de las grietas conductivas y algunas simulaciones preliminares también se realizan en tres dimensiones. Principales características del fenómeno de la propagación de la grieta predicho por las simulaciones de esta tesis se comparan directamente con las observaciones experimentales.

The investigation of failure in brittle materials, subjected to dynamic transient loading conditions, represents one of the ongoing challenges in the mechanics community. Progresses on this front are required to support the design of engineering components which are employed in applications involving extreme operational regimes. To this purpose, this thesis is devoted to the development of a framework which provides the capabilities to model how crack patterns form and evolve in brittle materials and how they affect the quantitative description of failure. The proposed model is developed within the context of diffusive interfaces which are at the basis of a new class of theories named phase field models. In this work, a set of additional features is proposed to expand their domain of applicability to the modelling of (i) rate and (ii) pressure dependent effects. The path towards the achievement of the first goal has been traced on the desire to account for micro-inertia effects associated with high rates of loading. Pressure dependency has been addressed by postulating a mode-of-failure transition law whose scaling depends upon the local material triaxiality. The governing equations have been derived within a thermodynamically-consistent framework supplemented by the employment of a micro-forces balance approach. The numerical implementation has been carried out within an updated lagrangian finite element scheme with explicit time integration. A series of benchmarks will be provided to appraise the model capabilities in predicting rate-pressure-dependent crack initiation and propagation. Results will be compared against experimental evidences which closely resemble the boundary value problems examined in this work. Concurrently, the design and optimization of a complimentary, improved, experimental characterization platform, based on the split Hopkinson pressure bar, will be presented as a mean for further validation and calibration.

The primary focus of this research is to evaluate the time-dependent crack growth in rocks using lab tests and numerical modeling and its application to geologic hazard problems. This research utilized Coconino sandstone and Columbia granite as the study materials and produced the subcritical crack growth parameters in both mode I and II loadings using the rock materials. The mode I loading test employs three different types of fracture mechanics tests: the Double Torsion (DT), the Wedge Splitting (WS), and the Double Cantilever Beam (DCB) test. Each test measured the mode I crack velocity. The DT test indirectly measured the crack velocity using the load relaxation method. The WS and DCB tests directly measured the crack velocity by monitoring using a video recording. The different mode I subcritical crack growth parameters obtained from the three tests are discussed. For the mode II loading test, this study developed a new shear fracture toughness test called the modified Punch-Through Shear (MPTS). The MPTS test conducted at different loading rates produced the mode II subcritical crack growth parameters. These fracture mechanics tests were calibrated and simulated using the distinct element method (DEM) and the finite element method (FEM). DEM analysis employed the particle flow code (PFC) to simulate the mixed mode crack growth and to match with the failure strength envelop of the triaxial compressive tests. FEM analysis employed the Phase2 program to analyze the crack tip stress distribution and the FRANC2D program to calculate the modes I and II stress intensity factors. The fracture mechanics tests and numerical modeling showed well the dependency of the mode II subcritical crack growth parameters according to confining pressure, loading rate, and the mode II fracture toughness. Finally, the UDEC modeling based on DEM is utilized in this study to forecast the long-term stability of the Coconino rock slope, as one of geologic hazards. The fracture mechanics approach is implemented in the program using the modes I and II subcritical crack growth parameters obtained from the lab tests and numerical modeling. Considering the progressive failure of rock bridges due to subcritical crack growth, the UDEC results predicted the stable condition of the Coconino rock cliff over 10,000 years. This result was validated by comparing it with the previous planar failure case.