Phase Field Modeling Of Fracture Research Articles

Phase field fracture in elasto-plastic solids: a length-scale insensitive model for quasi-brittle materials

Phase-field methods for fracture have been integrated with plasticity for better describing constitutive behaviours. In most of the previous phase-field models, however, the length-scale parameter must be interpreted as a material property in order to match the material strength in experiments. This study presents a phase-field model for fracture coupled with plasticity for quasi-brittle materials with emphasis on insensitivity of the length-scale parameter. The proposed model is formulated using variational principles and implemented numerically in the finite element framework. The effective yield stress is calibrated to vary with the length-scale parameter such that the tensile strength remains the same. Moreover, semi-analytical solutions are derived to demonstrate that the length-scale parameter has a negligible effect on the stress–displacement curve. Five representative examples are considered here to validate the phase-field model for fracture in quasi-brittle materials. The simulated force–displacement curves and crack paths agree well with the corresponding experimental results. Importantly, it is found that the global structural response is insensitive to the length scale though it may influence the size of the failure zone. In most cases, a large length-scale parameter can be used for saving the computational cost by allowing the use of a coarse mesh. On the other hand, a sufficiently small length-scale parameter can be selected to prevent overly diffusive damage, making it possible for the proposed phase-field model to simulate the fracture behaviour with $$ \varGamma $$ -convergence.

Phase-field modeling of fracture and crack growth in friction stir processed pure copper

The present work aims to predict the fracture mechanism of a friction stir processed (FSPed) copper. In particular, a phase-field model integrated with finite element model is developed for ductile fracture modeling. The phase-field evolution governed by the nonlinear coupled system comprising the linear momentum equation and the diffusion-type equation is solved concurrently through a Newton–Raphson approach. The proposed fracture model is established by correlating the phase-field degradation function with a scalar measure of the plastic strain, and assuming that the fracture takes place once the accumulated plastic strain reaches a critical value. The numerical simulation results are validated with experimental investigations. The results show that the proposed model is capable of capturing the experimentally observed sequence of elastoplastic base material behavior, FSP deformation, and fracture phenomena in specimens.

Phase field simulation for fracture behavior of hyperelastic material at large deformation based on edge-based smoothed finite element method

Phase Field Method (PFM) has remarkable performance in simulating crack patterns with complex topology and the edge-based smoothed finite element method (ES-FEM) has intrinsic advantages in modeling the fracture behavior at large deformation. A phase field fracture model for modeling hyperelastic material at large deformation is proposed based on ES-FEM. The weak forms and numerical framework of proposed model are thoroughly derived and constructed. The Neo-Hooken and Mooney-Rivlin constitutive models are adopted to describe the large deformation response of rubber-like materials under displacement load. Fracture behaviors of the penny shaped pre-cracked specimen and the double edge notched specimen in tension are simulated. The load-displacement curves of the double edge notched specimen agree well with that of the experiments. The influence of material constitutive model on Mode II crack propagation is investigated. It is shown that the developed model can capture the Mode I and the mode II cracks of rubber-like material and is insensitive to the distorted mesh at large deformation.

A phase-field model of frictional shear fracture in geologic materials

Geologic shear fractures such as faults and slip surfaces involve marked friction along the discontinuities as they are subjected to significant confining pressures. This friction plays a critical role in the growth of these shear fractures, as revealed by the fracture mechanics theory of Palmer and Rice decades ago. In this paper, we develop a novel phase-field model of shear fracture in pressure-sensitive geomaterials, honoring the role of friction in the fracture propagation mechanism. Building on a recently proposed phase-field method for frictional interfaces, we formulate a set of governing equations for different contact conditions (or lack thereof) in which frictional energy dissipation emerges in the crack driving force during slip. We then derive the degradation function and the threshold fracture energy of the phase-field model such that the stress–strain behavior is insensitive to the length parameter for phase-field regularization. This derivation procedure extends a methodology used in recent phase-field models of cohesive tensile fracture to shear fracture in frictional materials in which peak and residual strengths coexist and evolve by confining pressure. The resulting phase-field formulation is demonstrably consistent with the theory of Palmer and Rice. Numerical examples showcase that the proposed phase-field model is a physically sound and numerically efficient method for simulating shear fracture processes in geologic materials, such as faulting and slip surface growth.

A phase field modeling approach of cyclic fatigue crack growth

Phase field modeling of fracture has been in the focus of research for over a decade now. The field has gained attention properly due to its benefiting features for the numerical simulations even for complex crack problems. The framework was so far applied to quasi static and dynamic fracture for brittle as well as for ductile materials with isotropic and also with anisotropic fracture resistance. However, fracture due to cyclic mechanical fatigue, which is a very important phenomenon regarding a safe, durable and also economical design of structures, is considered only recently in terms of phase field modeling. While in first phase field models the material’s fracture toughness becomes degraded to simulate fatigue crack growth, we present an alternative method within this work, where the driving force for the fatigue mechanism increases due to cyclic loading. This new contribution is governed by the evolution of fatigue damage, which can be approximated by a linear law, namely the Miner’s rule, for damage accumulation. The proposed model is able to predict nucleation as well as growth of a fatigue crack. Furthermore, by an assessment of crack growth rates obtained from several numerical simulations by a conventional approach for the description of fatigue crack growth, it is shown that the presented model is able to predict realistic behavior.

Irreversibility and alternate minimization in phase field fracture: a viscosity approach

This work is devoted to the analysis of convergence of an alternate (staggered) minimization algorithm in the framework of phase field models of fracture. The energy of the system is characterized by a nonlinear splitting of tensile and compressive strains, featuring non-interpenetration of the fracture lips. The alternating scheme is coupled with an L^{2}-penalization in the phase field variable, driven by a viscous parameter delta >0, and with an irreversibility constraint, forcing the monotonicity of the phase field only w.r.t. time, but not along the whole iterative minimization. We show first the convergence of such a scheme to a viscous evolution for delta >0 and then consider the vanishing viscosity limit delta rightarrow 0.

A phase-field model for mixed-mode fracture based on a unified tensile fracture criterion

Different fracture patterns can be observed because of different material properties, even the geometry and loading are the same. However, most of the known phase-field fracture models have only considered the tensile failure and may not be directly applicable to the shear fracture. In this paper, a phase-field model for mixed-mode fracture is proposed based on a unified tensile fracture criterion. The proposed model is developed from the unified phase-field theory and the original unified phase-field model can be recovered as a particular case. General softening laws for cohesive zone models can also be considered. The unified tensile fracture criterion is embedded in the proposed mixed-mode phase-field model and different fracture patterns can be obtained in the simulation according to the material properties, including failures based on both maximum normal stress and maximum shear stress criteria. The crack propagation direction can be easily determined by the unified tensile fracture criterion. Compared with the classical phase-field model, two additional material parameters are needed, i.e., the failure tension strength and the ratio of the critical shear failure stress to the critical normal fracture stress. Numerical examples have shown that the proposed model has the ability to model mixed-mode fractures, and can also be applied to rock-like brittle materials under compression.

Parallel Matrix-Free Higher-Order Finite Element Solvers for Phase-Field Fracture Problems

Phase-field fracture models lead to variational problems that can be written as a coupled variational equality and inequality system. Numerically, such problems can be treated with Galerkin finite elements and primal-dual active set methods. Specifically, low-order and high-order finite elements may be employed, where, for the latter, only few studies exist to date. The most time-consuming part in the discrete version of the primal-dual active set (semi-smooth Newton) algorithm consists in the solutions of changing linear systems arising at each semi-smooth Newton step. We propose a new parallel matrix-free monolithic multigrid preconditioner for these systems. We provide two numerical tests, and discuss the performance of the parallel solver proposed in the paper. Furthermore, we compare our new preconditioner with a block-AMG preconditioner available in the literature.

A hybrid phase field model for fracture induced by lithium diffusion in electrode particles of Li-ion batteries

Lithium-ion batteries (LIBs) of high energy density and light-weight design, find wide applications in electronic devices and systems. Degradation mechanisms that caused by lithiation is a main challenging problem for LIBs with high capacity electrodes like silicon (Si), which eventually can reduce the lifetime of batteries. In this paper, a hybrid phase field model (PFM) is proposed to study the fracture behavior of LIB electrodes. The model considers the coupling effects between lithium (Li) -ion diffusion process, stress evolution and crack propagation. Also, the dependency of Elastic properties on the concentration magnitude of Li-ion is considered. A numerical implementation based on a MATLAB finite element (FE) code is elaborated. Then, the proposed hybrid PF approach is applied to a Nanowire (NW) Si electrode particle. Numerical results show that the hybrid model shows less tendency to crack growth than the isotropic model.

Evaluation of variational phase-field models for dynamic brittle fracture

During the last decades various phase-field model have been developed in the variational approach to model dynamic fracture of brittle solids. Complex fracture phenomena such as crack branching, dynamic crack instability and fragmentation, etc., can be captured with a minimal list of physical parameters. However, an objective evaluation of existing results is missing and in particular, the issues of mesh size dependence and length scale sensitivity have been rarely studied. Aiming to bridge the above gap, this paper presents an extensive evaluation of commonly used variational phase-field models for dynamic fracture to address the question: how well can we predict dynamic fracture with current PFMs. Several popular PFMs in the literature are considered, implemented within a single code platform and used to simulate six dynamic fracture problems of homogeneous brittle solids. Mesh convergence and length scale sensitivity are systematically analyzed. Quantitative evaluation of the branching angle, crack tip position and/or velocity with respect to available experiments, and comparison against the results of other approaches, are both presented.

A phase-field model of fracture with frictionless contact and random fracture properties: Application to thin-film fracture and soil desiccation

We present a new derivation for a phase-field model of cohesive fracture that allows for fully-damaged surfaces to properly transmit tractions under frictionless contact conditions. The model is derived from an energy minimization standpoint, and the governing equations are presented in a general Allen–Cahn form, unifying both brittle and cohesive fracture models. A novel elastic energy split is proposed to enforce frictionless contact conditions along the regularized crack set. A fixed-point iterative algorithm is used to solve the system of equations, and an active-set strategy is incorporated to enforce the crack set irreversibility. The model is then applied to study the mechanical-fracture coupling in the fracture of thin films and soil desiccation problems in which intricate crack networks form. We present results for one-dimensional, two-dimensional, and fully three-dimensional configurations, and examine the relationship between the generalized driving energies for fracture and the resulting fracture patterns. We also investigate the impact of stochastic spatial variations in material parameters on fragment statistics and fracture network morphology. To this end, we develop a random field model for fracture properties and address a forward propagation problem. The versatility offered by the probabilistic framework is finally highlighted by considering a statistical inverse problem based on physical experiments. It is shown, in particular, that the formulation allows key characteristics of observed crack patterns to be reproduced with reasonable accuracy.

On crack opening computation in variational phase-field models for fracture

Phase-field models for fracture have gained exceptional popularity in the last couple of decades and have been extended into areas well beyond brittle quasi-static fracture propagation to ductile, dynamic, or hydraulic fracturing. Despite the significant theoretical advancements in these more complex physical settings, little attention has been paid to the quantification of crack opening displacement to date as most applications do not explicitly require the crack opening displacement for the morphological evolution of cracks. However, one of the exemptions would be hydraulic fracturing where the crack propagation is driven by the fluid pressure which strongly depends on the crack opening displacement.In this study, we look into two known approaches, a line integral and a level-set method, for crack opening computation mainly from an implementation point of view. Firstly, we derive an approximation of a discontinuous function field in the variational phase-field setting which is then applied to obtain the crack opening (displacement jump) and verify the approximation against a closed solution. We then propose a “certain distance from the crack” required for the level-set function using a one-dimensional analysis. Finally, we compare these approaches under several different conditions such as crack alignment to the mesh or under loading (asymmetric displacement field) and investigate the convergence with respect to the mesh size.

A phase field model for cohesive fracture in micropolar continua

While crack nucleation and propagation in the brittle or quasi-brittle regime can be predicted via variational or material-force-based phase field fracture models, these models often assume that the underlying elastic response of the material is not size-dependent. Yet, a length scale parameter must be introduced to these models to enable sharp cracks properly represented by a regularized implicit function. However, many materials with internal microstructures that contain surface tension, micro-cracks, micro-fracture, inclusion, cavity or those of particulate nature often exhibit size-dependent behaviors in both the path-independent and path-dependent regimes. This paper is intended to introduce a unified treatment that captures the size effect of the materials in both elastic and damaged states. By introducing a cohesive micropolar phase field fracture theory, along with the computational model and validation exercises, we explore the interacting size-dependent elastic deformation and fracture mechanisms exhibits in materials of complex microstructures. To achieve this goal, we introduce the distinctive degradation functions of the force-stress–strain and couple-stress-micro-rotation energy-conjugated pairs for a given regularization profile such that the macroscopic size-dependent responses of the micropolar continua are insensitive to the length scale parameter of the regularized interface. Then, we apply the variational principle to derive governing equations from the micropolar stored energy and dissipative functionals. Numerical examples are introduced to demonstrate the proper way to identify material parameters and the capacity of the new formulation to simulate complex crack patterns in the quasi-static regime.

Non-conforming multipatches for NURBS-based finite element analysis of higher-order phase-field models for brittle fracture

This paper proposes an effective computational tool for brittle crack propagation problems based on a combination of a higher-order phase-field model and a non-conforming mesh using a NURBS-based isogeometric approach. This combination, as demonstrated in this paper, is of great benefit in reducing the computational cost of using a local refinement mesh and a higher-order phase-field, which needs higher derivatives of basis functions. Compared with other approaches using a local refinement mesh, the Virtual Uncommon-Knot-Inserted Master-Slave (VUKIMS) method presented here is not only simple to implement but can also reduce the variable numbers. VUKIMS is an outstanding choice in order to establish a local refinement mesh, i.e. a non-conforming mesh, in a multi-patch problem. A phase-field model is an efficient approach for various complicated crack patterns, including those with or without an initial crack path, curved cracks, crack coalescence, and crack propagation through holes. The paper demonstrates that cubic NURBS elements are ideal for balancing the computational cost and the accuracy because they can produce accurate solutions by utilising a lower degree of freedom number than an extremely fine mesh of first-order B-spline elements.

Numerical modelling and simulations on the mechanical failure of bulk superconductors during magnetization: based on the phase-field method

Mechanical failure is one of the most significant challenges for the application of bulk superconductors acting as trapped field magnets. In this paper, a numerical simulation framework based on the coupled H-formulation for electromagnetics of superconductors and the phase-field fracture model for solids is proposed and implemented to simulate the crack nucleation and propagation within bulk superconductors during the magnetization process. The thermal stress due to cool down and the huge Lorentz force during the magnetization process are considered in the simulation. The magnetic field, current density and stress/strain field distribution, and the crack initiation and propagation within the homogeneous and inhomogeneous bulk sample under high magnetic field are numerically simulated, respectively. The trapped magnetic fields for the bulk superconductors under different magnetic fields are obtained numerically. In addition, the influences of microstructures such as voids, pre-existing microcracks on the trapped magnetic field of the bulk superconductor are also investigated and presented, which demonstrate the flexibility of the proposed framework to investigate the mechanical failure of bulk superconductors.

A higher order nonlocal operator method for solving partial differential equations

A higher order nonlocal operator method for the solution of boundary value problems is developed. The proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method (Ren et al., 2020) which only ensures first-order convergence. Furthermore, it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions. Only the functionals based on the nonlocal operators (termed as operator functional) are needed to obtain the final discrete system of equations, which significantly facilitates the implementation. Several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the Poisson equation in 2–5 dimensional space, Kirchhoff and von Kármán plate problems, incompressible elastic materials as well as phase field modeling of fracture.

Correlation between microstructural characteristics from micro-CT of foamed concrete and mechanical behaviors evaluated by experiments and simulations

Microstructural characteristics and properties of foamed concrete, a cellular lightweight cement paste with entrained air voids, are investigated. The complex 3D microstructures of the foamed concrete with different densities are obtained by micro-CT imaging and binarized as two-phase (pore/solid) materials by image processing. The mechanical responses of the microstructures are evaluated by simulations within the phase-field fracture model, and the input material parameters for simulations are identified. Microstructural characteristics and properties from experiments and simulations are correlated. In particular, linear attenuation coefficients for the solid phase in micro-CT images correlate with input material parameters. These correlations can serve to determine solid-phase input material parameters in foamed concretes with different densities. The behavior of foamed-concrete binders, modeled as multi-phase solids, should thus be evaluated more accurately through simulations, based on the observed correlations.

Phase field modeling of Hertzian indentation fracture

The formation and subsequent growth of cracks in brittle solids caused by indentation loading is investigated using a phase field modeling of fracture. While this methodology, at first glance, appears to be a promising tool for handling problems like the present one including crack initiation from an initially defect-free surface, its actual application reveals a number of critical issues. The paper illustrates inherent shortcomings of common phase field models for fracture based on a variational structure. Appropriate modifications are discussed which render the method – at least to some extent – suitable for the simulation of indentation fracture. Particular emphasis is placed on the significance of the material’s tensile strength and fracture toughness as independent physical properties necessary for the quantitative description of crack formation. Numerical predictions are compared with other theoretical results as well as experimental findings from the literature.

A multi phase-field fracture model for long fiber reinforced composites based on the Puck theory of failure

Phase-Field (PF) methods of fracture have emerged as powerful modeling tools for triggering fracture events in solids. These numerical techniques efficiently alleviate mesh dependent pathologies and are very suitable for characterizing brittle as well as quasi-brittle fracture in a wide range of engineering materials and structures including fiber reinforced composites. In this work, a multi phase-field model relying on the Puck’s failure theory is proposed for triggering intra-laminar cracking in long fiber reinforced composites. The current formulation encompasses the differentiation of fiber and inter-fiber (matrix-dominated) failure phenomena via the consideration of two independent phase-field damage-like variables, and the corresponding evolution equations and length scales. Moreover, for matrix-dominated deformation states, the present formulations endow the incorporation of plastic effects via an invariant-based plasticity model. Special attention is also devoted to its finite element implementation, which is conducted using the user-defined capabilities UMAT and UEL of ABAQUS, in conjunction with the thorough assessment of its thermodynamic consistency. Several representative applications pinpoint the applicability of the proposed computational tool.

A statistical geometry approach to length scales in phase field modelling of fracture and strength of porous microstructures

In phase field methods for fracture, versatility is acquired at the cost of the addition of a new parameter, a length scale parameter. The length scale parameter affects notch sensitivity, which in cellular materials is typically related to lengths of the material microstructure. Here, the relation between this length scale parameter and observable microstructural lengths of a cellular material is investigated numerically, specifically lengths derived using statistical geometry of random Voronoi tessellations.It is found that the fracture load of a homogeneous continuum model (i.e. a macroscopic model) coincides with that of a microstructured model if the length scale parameter is chosen to be the same in both models, while approximate macroscale stiffness and energy release rate are obtained by scaling the properties of the microstructured model with powers of the relative density. The correlation between the micro- and macroscale models is best when the length parameter is chosen as approximately two to three times the average cell size of the microstructure, depending on the relative density – which is also equal to approximately eight times a critical defect length of the Voronoi tessellation, regardless of relative density – as the microstructured material then behaves more like a continuum. If the length scale parameter needs to be smaller than twice the cell size or five times the critical length, the crack path is sensitive to features in the microstructure, and continuum modelling of the porous material cannot be advised.