Phase Field Modeling Of Fracture Research Articles

Smoothed point interpolation methods for phase-field modelling of pressurised fracture

The problem of hydraulic fracturing is of great relevance to various areas and is characterised by the occurrence of complex crack patterns with bifurcations and branches. For this reason, an interesting approach is the modelling of hydraulic fracture using a phase-field model. In addition to the discretisation using the Finite Element Method (FEM), some works have already explored the discretisation of the phase-field model with meshfree methods, including the Smoothed Point Interpolation Methods (SPIM) family. Seeking to take advantage of the good convergence results of SPIM for phase-field modelling of brittle fractures, this paper proposes the use of SPIM for phase-field modelling of pressurised fractures. In order to limit the computational cost, a prescribed SPIM-FEM coupling is employed, with the purpose of concentrating the meshless discretisation only in the regions of expected crack propagation. The model is characterised by a constant internal pressure load along the fracture that is applied indirectly from the formulation of the phase-field model. A series of numerical simulations is presented. The aim is to evaluate the proposed model, verify the results and point out characteristics of the phase-field model with internal pressure.

What is the physical origin of the gradient flow structure of variational fracture models?

We investigate a physical characterization of the gradient flow structure of variational fracture models for brittle materials: a Griffith-type fracture model and an irreversible fracture phase field model. We derive the Griffith-type fracture model by assuming that the fracture energy in Griffith's theory is an increasing function of the crack tip velocity. Such a velocity dependence of the fracture energy is typically observed in polymers. We also prove an energy dissipation identity of the Griffith-type fracture model, in other words, its gradient flow structure. On the other hand, the irreversible fracture phase field model is derived as a unidirectional gradient flow of a regularized total energy. We have considered the time relaxation parameter a mathematical approximation parameter, which we should choose as small as possible. In this research, however, we reveal the physical origin of the gradient flow structure of the fracture phase field model (F-PFM) and show that the small time relaxation parameter is characterized as the rate of velocity dependence of the fracture energy. It is verified by comparing the energy dissipation properties of those two models and by analysing a travelling wave solution of the irreversible F-PFM. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Cracking mechanism of CMAS-corroded thermal barrier coatings based on a coupled thermo-chemo-mechanically phase-field fracture model

A thermo-chemo-mechanically coupled theoretical framework is proposed to describe the calcium-magnesium-alumina-silicate (CMAS) corrosion process during the cooling process. A phase-field fracture model is developed to investigate the effect of cooling temperature and CMAS concentration on the degree of corrosion reaction, the stress evolution and the crack initiation and propagation. σ11 concentrates in the region beneath the overlay of CMAS and σ22 appears at the interface between top ceramic coating (TC) and bond coating (BC). The higher stress concentration of σ11 and σ22 contribute to the formation of both vertical and transverse cracks. Transverse cracks first emerge at the interface between TC and BC in the edge region, followed by the formation of vertical cracks in the CMAS-coated region. Vertical cracks propagate to the interface and deflect into transverse cracks. The transverse cracks at the interface further propagate and merge, ultimately leading to the coating delamination. The higher initial cooling temperature and CMAS concentration contribute to the accelerated development of vertical cracking and the increase of the quantity and length of transverse and vertical cracks. The model provides a significant advantage in predicting the failure of TBCs during the cooling stage of CMAS corrosion.

Phase-field modeling of thermal shock fracture in functionally graded materials

In this paper, the thermo-elastic fracture problems of functionally graded materials (FGMs) are thoroughly investigated based on a modified phase field model. In this model, the material constants and fracture toughness vary along a specified direction, the thermal conductivity and stiffness constants in the damaged regions are degraded by the phase-field variable, and the crack evolution is driven by the variation of elastic energy induced by the thermo-mechanical loading. Therefore, the temperature, mechanical and damage fields are coupled with each other. The validation of the present approach has been checked by comparing the present results with the analytical, numerical, and experimental results in literature. The examples of thermo-elastic tensile fracture and thermal shock cracking of FGMs are performed to investigate the influence of material heterogeneity and external thermal loading. Furthermore, the experimental phenomenon of crack deflection in FGMs has been captured by the present simulations. The present study provides a guidance for understanding the fracture mechanism, material design and safety assessment of FGM structures in engineering practice.

Mechanical property evaluation of 3D multi-phase cement paste microstructures reconstructed using generative adversarial networks

This study proposes an artificial intelligence based framework for reconstructing the 3D multi-phase cement paste microstructure to evaluate its mechanical properties using simulation. The reconstruction of cement paste microstructures is performed using modified generative adversarial networks (GANs) based on microstructural images from micro-CT. For computational efficiency, 2D microstructures are first reconstructed and then extended to 3D microstructures. The reconstructed microstructures exhibit the same microstructural features as the original microstructures when characterized by probability functions. Mechanical properties such as stiffness and tensile strength are evaluated for the original and reconstructed specimens using a phase-field fracture model, and similar behaviors are observed. The results confirm that the reconstructed virtual microstructures can be used to supplement the real microstructures in evaluating the mechanical properties of 3D multi-phase cement paste. This approach thus provides a critical element of a data-driven approach to correlating its microstructure and properties.

Modelling and simulation of natural hydraulic fracturing applied to experiments on natural sandstone cores

Under in-situ conditions, natural hydraulic fractures (NHF) can occur in permeable rock structures as a result of a rapid decrease of pore water accompanied by a local pressure regression. Obviously, these phenomena are of great interest for the geo-engineering community, as for instance in the framework of mining technologies. Compared to induced hydraulic fractures, NHF do not evolve under an increasing pore pressure resulting from pressing a fracking fluid in the underground but occur and evolve under local pore-pressure reductions resulting in tensile stresses in the rock material. The present contribution concerns the question under what quantitative circumstances NHF emerge and evolve. By this means, the novelty of this article results from the combination of numerical investigations based on the Theory of Porous Media with a tailored experimental protocol applied to saturated porous sandstone cylinders. The numerical investigations include both pre-existing and evolving fractures described by use of an embedded phase-field fracture model. Based on this procedure, representative mechanical and hydraulic loading scenarios are simulated that are in line with experimental investigations on low-permeable sandstone cylinders accomplished in the Porous Media Lab of the University of Stuttgart. The values of two parameters, the hydraulic conductivity of the sandstone and the critical energy release rate of the fracture model, have turned out essential for the occurrence of tensile fractures in the sandstone cores, where the latter is quantitatively estimated by a comparison of experimental and numerical results. This parameter can be taken as reference for further studies of in-situ NHF phenomena and experimental results.

A micromagnetic-mechanically coupled phase-field model for fracture and fatigue of magnetostrictive alloys

Magnetostrictive alloys are usually brittle materials with micromagnetic structures. Their structural reliability and durability depend on the complex micromagnetic-mechanical coupling at smaller length scales encompassing the evolution of micromagnetic structures. Herein we propose a micromagnetic-mechanically coupled phase-field model for fracture and fatigue behavior of magnetostrictive alloys with evolution of the micromagnetic structure. The thermodynamically-consistent model is derived from microforce theory, laws of thermodynamics, and Coleman–Noll analysis. The evolution of crack phase-field and magnetization-vector order parameters that are fully coupled is governed by history field dependent Allen–Cahn and Landau–Lifshitz–Gilbert equations, respectively. The model is extended to fatigue by introducing a degradation prefactor for the fracture energy as a function of positive elastic energy. One-dimensional analyses are then presented to anatomize the crack driving forces in terms of fully coupled micromagnetic-mechanical and pure mechanical driving force. We demonstrate the model capabilities by finite-element numerical studies on the micromagnetic domain evolution during the crack propagation and the influence of external magnetic field for type-I, type-II, and three-point bending fracture, as well as for the fracture of a single-edge notched specimen with an elliptical inclusion. The simulation result shows that depending on how micromagnetic domains are switched under micromagnetic-mechanical coupling, the magnetic field can enhance or decrease the critical load. In the presence of inclusion with larger fracture toughness, a crack is found to nucleate in the tri-junction of multi-domain micromagnetic structure owing to the high elastic strain around the tri-junction point. It is further found that a suitable magnetic field promoting magnetization-vector rotation around the crack tip could remarkably improve the fracturing load and fatigue life. The results demonstrate the model promising for the study of micromagnetic-mechanically coupled fracture and fatigue in magnetostrictive alloys.

Phase-field ductile fracture simulations of thermal cracking in additive manufacturing

We present a multiphysics phase-field fracture model for thermo-elasto-plastic solids in the context of finite deformation and apply it to simulate the hot cracking phenomenon during metal additive manufacturing. The model is derived in a thermodynamically consistent manner, with the intercoupling mechanisms among elastoplasticity, phase-field crack and heat transfer comprehensively considered. It involves particularly coupled parameters among these materials physics, e.g. plasticity-dependent degradation function and fracture toughness, damage-dependent yield surface and thermal properties, and temperature-dependent elastoplastic properties and fracture strength. The finite element implementation of the coupled phase-field model is benchmarked with simulation results of a tensile test of an I-shape specimen, encompassing elastoplasticity, hardening, necking, crack initiation and propagation, in contrast to the related experimental results. The validated model is further employed to simulate the multiphysics hot cracking phenomenon in additive manufacturing in the context of both the effective powder-bed model and the powder-resolved model thanks to prior non-isothermal phase-field powder-bed-fusion simulations. Simulation results reveal certain key features of the hot crack and its dependency on process parameters like beam power and scan speed, which are helpful for the fundamental understanding of crack formation mechanisms and process optimization.

A phase-field fracture model in thermo-poro-elastic media with micromechanical strain energy degradation

This work extends the hydro-mechanical phase-field fracture model to non-isothermal conditions with micromechanics based poroelasticity, which degrades Biot’s coefficient not only with the phase-field variable (damage) but also with the energy decomposition scheme. Furthermore, we propose a new approach to update porosity solely determined by the strain change rather than damage evolution as in the existing models. As such, these poroelastic behaviors of Biot’s coefficient and the porosity dictate Biot’s modulus and the thermal expansion coefficient. For numerical implementation, we employ an isotropic diffusion method to stabilize the advection-dominated heat flux and adapt the fixed stress split method to account for the thermal stress. We verify our model against a series of analytical solutions such as Terzaghi’s consolidation, thermal consolidation, and the plane strain hydraulic fracture propagation, known as the KGD fracture. Finally, numerical experiments demonstrate the effectiveness of the stabilization method and intricate thermo-hydro-mechanical interactions during hydraulic fracturing with and without a pre-existing weak interface.

Phase-field modeling of fracture with physics-informed deep learning

We explore the potential of the deep Ritz method to learn complex fracture processes such as quasistatic crack nucleation, propagation, kinking, branching, and coalescence within the unified variational framework of phase-field modeling of brittle fracture. We elucidate the challenges related to the neural-network-based approximation of the energy landscape, and the ability of an optimization approach to reach the correct energy minimum, and we discuss the choices in the construction and training of the neural network which prove to be critical to accurately and efficiently capture all the relevant fracture phenomena. The developed method is applied to several benchmark problems and the results are shown to be in qualitative and quantitative agreement with the finite element solution. The robustness of the approach is tested by using neural networks with different initializations.

Phase Field Fracture Model for Assessing the Load Bearing Capacity of Fractured Glass

The use of glass as structural material has highlighted the need for more reliable numerical approaches to analyze its mechanical behavior, especially in the accidental eventuality of fracture. Modelling the behavior of fractured laminated glass, in fact, is fundamental to assess the Post-Fracture load-bearing capacity. However, this is a highly challenging task because of the many interplaying factors, such as the viscoelastic and thermal-dependent behavior of the interlayer, the presence of a highly complex and variable crack pattern and the interaction among fragments. The objective of the present work is the development and testing of a robust numerical model that can naturally introduce the generated crack pattern into virtual specimens and manage the interaction among many fragments. The phase field fracture model is herein explored, by assigning the damage variable to fit the pre-existing crack pattern. Then, the specimen is loaded letting the phase field managing the fragments interaction. The dependence of the stress tensor with the damage variable is herein defined through the Cleavage-Deviatoric model, since it prevents fully damaged regions from transmitting tensile and shear stresses yet keeping their ability to bear compressive forces. Indeed, this model can asymptotically reproduce unilateral and frictionless contact conditions between the existing crack lips. Preliminary case studies are discussed to check the potentiality of the proposed approach.

Length Scale Insensitive Phase-Field Fracture Methodology for Brittle and Ductile Materials

We present length scale insensitive phase-field fracture models for brittle and ductile fracture to address the deficiencies of widely implemented models which over-estimate crack dissipation. An approach is proposed to attain a regularization length scale insensitive mechanical response, which considers a continuous approximation of a crack boundary with a function of infinite support. This is in contrast to previous approaches to attain a length scale insensitive response which exploited approximations of the crack boundary with functions of finite support. The choice of these functions with finite support creates implementational challenges which are avoided in the presented quasi-brittle and ductile fracture models with infinite support of the phase-field. The capability of these models is demonstrated in several benchmarks with attention given to the sensitivity of the structural response with respect to the choice of regularization length scale. The models are validated against a three-point bending test on a concrete specimen and an in-plane shear test on a steel specimen.

Domain decomposition methods and acceleration techniques for the phase field fracture staggered solver

AbstractThe phase field modeling of fracture is able to simulate the nucleation and the propagation of complex crack patterns. However, the relatively small internal lengths that are required usually lead to very fine meshes and high computational costs, especially for three‐dimensional applications. In the present work, additional cost also comes from the implicit dynamics regularization of unstable crack propagations which potentially leads to a large variation of time steps when switching from quasi‐static to dynamic regimes. To reduce the time to solution in this context, this study proposes a domain decomposition framework and acceleration techniques for the phase field fracture staggered solver. The displacement subproblem and the phase field one are solved with parallel domain decomposition solvers. Dual domain decomposition methods provide low cost preconditioner well adapted to the phase field subproblem. For displacement subproblems undergoing unstable crack propagations, primal domain decomposition methods are preferred to be less sensitive to the treatment of floating substructures. Preconditioners performances are assessed and scalability studies over academic test cases, up to 324 subdomains, are presented. Finally, the robustness of the approach is illustrated on two semi‐industrial simulations.

A phase-field fracture model for creep-fatigue behavior

A phase-field fracture model for creep-fatigue behavior is proposed. The fatigue and creep behavior are simulated by modifying the free energy functional through a total degradation function that diminishes fracture toughness with the accumulation of the relevant history variable. Specifically, strain energy density and creep strain serve as the chosen fatigue and creep history variables, respectively. To model the creep-fatigue interaction, we propose several total degradation function formulations comprising fatigue and creep degradation functions. Numerical experiments demonstrate the capability of our model to reproduce the strain-lifetime relationship and crack growth behavior under creep-fatigue loading conditions with varying hold times. Moreover, we simulated examples of plates with holes, demonstrating the capability of our model for complex crack paths.

Phase-field model for 2D cohesive-frictional shear fracture: An energetic formulation

This paper presents a phase-field model for cohesive-frictional shear fracture. The model is derived based on two energetic principles: the energy conservation law and a variational inequality of virtual work that serves as a stability condition. We show that all the governing equations for frictional fracture can be obtained from the above two principles, including the equilibrium condition, the phase-field evolution law and, most importantly, the yield function and flow rule for the plasticity-like frictional slip. The information of crack direction is naturally included in the flow rule. Theoretical and numerical results show that the proposed model is faithfully consistent with the Mohr–Coulomb theory of frictional failure in terms of the crack nucleation stress and direction. In addition, the presented phase-field model has an explicit frictional cohesive law.

A phase-field model for the brittle fracture of Euler–Bernoulli beams coupling stretching and bending

Damage gradient models seek to simulate fracture mechanics by modulating the material stiffness. Within this framework, a singular scalar field representing damage is commonly utilized to globally decrease elastic energy. However, when considering structural models like beams and plates, this approach often fails to adequately capture important aspects due to the interaction between stretching and bending contributions. In this work, we propose a model for planar Euler–Bernoulli beams that incorporates the following features: firstly, we utilize two phase-field damage parameters to describe material damaging, specifically addressing the ‘erosion’ occurring above and below the original, undamaged beam; secondly, we assume a simple linear dependence of the axial stress field on the thickness coordinate, along with linear dependence on the axial force and bending moment. By appropriately identifying the constitutive response, our model effectively considers the coupling between stretching and bending induced during through-the-thickness damage, demonstrating good agreement with 2D observations.

Modeling dynamic crack growth in quasicrystals: Unraveling the role of phonon–phason coupling

In this work, we employed the phase-field fracture (PFF) model for dynamic brittle fracture in quasicrystals (QCs). The dynamic PFF formulation is derived using both elastodynamics and elasto-hydrodynamic theory. The temporal discretization is achieved using an implicit generalized-α method, which is unconditionally stable for a particular choice of constants. The model incorporates two distinct crack driving forces: (a) the contribution of elastic strain energy from both phonon and phason fields and (b) the contribution solely from the elastic energy associated with the phonon field. To validate the implementation of QCs, we have conducted benchmarking against recent results in the literature, particularly in the context of quasi-static loading. We addressed various paradigmatic case studies to demonstrate the dynamic crack nucleation and propagation in planar 2D decagonal QCs, with a primary emphasis on understanding the interplay between phonon and phason fields. Phason walls, representing low-energy crack paths resulting from atomic rearrangements, play a significant role in crack propagation by introducing an additional energy component. Notably, increasing the phonon–phason coupling constant expedites both crack initiation and propagation. Furthermore, variations in the phonon–phason coupling parameter result in different crack trajectories, observed in both uniaxial and biaxial loading scenarios. The developed code can be downloaded from https://github.com/Hirshikesh/Quasicrystal.git.

Voronoi-based 3D phase field model and numerical simulations of granite crack propagation

The distribution of mineral grains within underground rocks and materials is crucial for understanding geological processes and the stability of engineering structures. However, numerical simulation methods that consider 3D mineral particles are still immature. This study presents a 3D phase-field fracture model of mineral grains based on Voronoi polygons, making some beneficial attempts in this regard. The phase-field method does not rely on mesh reconstruction or crack path tracking and can simulate complex crack propagation behavior. Meanwhile, the Voronoi polygon can better describe the microscopic structure of rock with grain boundaries. Rooted in Voronoi polygons, this model efficiently captures the complex geometric shapes of mineral grains and accurately characterizes their positions, shapes, and sizes. Therefore, this study combines Voronoi polygons to establish a 3D phase-field model. Through two numerical simulation cases, the model demonstrates its effectiveness in simulating the impact of three-dimensional mineral grains on crack propagation behavior, showing that cracks tend to propagate at the boundaries or inside softer mineral grains. The modeling approach proposed in this paper has good reference value for numerical simulation studies considering the microscopic structure of rocks.

On the energy decomposition in variational phase-field models for brittle fracture under multi-axial stress states

Phase-field models of brittle fracture are typically endowed with a decomposition of the elastic strain energy density in order to realistically describe fracture under multi-axial stress states. In this contribution, we identify the essential requirements for this decomposition to correctly describe both nucleation and propagation of cracks. Discussing the evolution of the elastic domains in the strain and stress spaces as damage evolves, we highlight the links between the nucleation and propagation conditions and the modulation of the elastic energy with the phase-field variable. In light of the identified requirements, we review some of the existing energy decompositions, showcasing their merits and limitations, and conclude that none of them is able to fulfil all requirements. As a partial remedy to this outcome, we propose a new energy decomposition, denoted as star-convex model, which involves a minimal modification of the volumetric-deviatoric decomposition. Predictions of the star-convex model are compared with those of the existing models with different numerical tests encompassing both nucleation and propagation.

Optimizing integration point density for exponential finite element shape functions for phase-field modeling of fracture in functionally graded materials

In the realm of fracture mechanics and phase-field modeling, the utilization of exponential finite element (EFE) shape functions has shown promise in accurately predicting fracture responses, particularly in scenarios with intricate crack propagation paths. However, the computational complexity associated with EFE shape functions, including higher integration schemes and orientation requirements, prompts the need for optimization. This study delves into the critical aspect of integration point density and aims to determine the ideal balance between accuracy and computational efficiency in the context of EFE shape functions. The research focuses on functionally graded materials, which inherently exhibit spatial variations in solution quantities due to their graded material system. Through a systematic investigation, the study aims to identify the optimal number of integration points required to maximize the computational advantages offered by EFE shape functions while minimizing the associated computational resources. By conducting a comprehensive analysis across various loading scenarios, such as tension and shear, this study intends to establish a quantitative relationship between integration point density and prediction accuracy for fracture responses.