Phase-field Modeling Of Brittle Fracture Research Articles

Phase-Field Modeling of Fused Silica Cone-Crack Vickers Indentation.

In this paper, a 3D phase-field model for brittle fracture is applied for analyzing the complex fracture patterns appearing during the Vickers indentation of fused silica. Although recent phase-field models for the fracture caused by the indentation loading have been verified by some simpler academic axis-symmetric examples, a proper validation of such models is still missing. In addition, heavy computational costs, and a complicated compression stress field under the indenter, which demands different energy decompositions, have been identified as the most important impediments for the successful application of the phase-field method for such problems. An adaptive strategy is utilized for reducing the computational costs, and some modifications are introduced, which enable an accurate simulation of the Vickers indentation fracture. Here, the fracture initiation ring outside the contact zone is detected by using different energy decompositions, and the dominant cone-crack formation under the Vickers indenter is observed. Different contact conditions are investigated. The proposed model is validated by experimental measurements, and a quantitative and qualitative comparison between experimental and numerical results is conducted.

An assessment of anisotropic phase-field models of brittle fracture

In several classes of ductile and brittle materials consisting of different cleavage planes, an orientation dependency of the fracture process is observed. It leads for instance to complex failure behaviours and crack paths in polycrystalline or architected materials. This paper focuses on modelling anisotropy of brittle fracture by means of a variational phase-field approach. More precisely, we study different models including several phase (or damage) variables corresponding to different damage mechanisms. First, we recall a multi-mechanism gradient damage model based on an anisotropic non-local fracture energy. We then consider a model accounting for an anisotropic degradation of the elasticity stiffness tensor. Both types of anisotropies are compared in terms of their influence on analytical homogeneous solutions under uniaxial and biaxial tensile loadings. Weak and strong anisotropies are captured via the chosen multi-mechanism damage framework. The models are implemented numerically by using a finite element discretization. In order to improve numerical performance, we implement an algorithm based on a hybrid direct–iterative resolution of the displacement sub-problem. Accuracy of model prediction is assessed by comparing numerical results to theoretical solutions under uniaxial loading. Benchmark numerical tests on notched and perforated plates highlight the role of material parameters on the fracture anisotropy. Furthermore, both models are able to retrieve zig-zag crack patterns observed in prior numerical and experimental studies. Finally, we discuss the predictions of a model combining both types of anisotropies.

Crack nucleation and propagation in the phase-field cohesive zone model with application to Hertzian indentation fracture

Hertzian indentation fracture (Hertz, 1896) is an intriguing problem with no pre-defects. Recent studies indicated that those popular phase-field models for brittle fracture might be restrictive in dealing with this problem — crack nucleation can be considered only for the failure strength within a rather limited range. In this work, this problem is analyzed by the phase-field cohesive zonde model (PF-CZM), focusing on the whole failure process of crack nucleation and propagation. Analytical and numerical results show that for any realistic failure strength, the ‘spontaneous’ crack nucleation in an initially defect-free surface, the subsequent small but rapid vertical extension and finally the stable cone-shaped crack propagation, etc., can all be qualitatively and quantitatively captured by the PF-CZM with no modification. This competence is credited to treating the failure strength and the fracture energy (toughness) as two independent material properties. Accordingly, the phase-field length scale is a numerical parameter of no further constraint, which can be made as small as possible such that the convergence to the Barenblatt (1959) cohesive zone model is guaranteed even for short cracks. Moreover, the seamless incorporation of the strength-based crack nucleation criterion and the energy-based crack propagation criterion endows the PF-CZM with the capability of tackling crack nucleation and propagation in pristine solids.

An open-source implementation of a phase-field model for brittle fracture using Gridap in Julia

This article proposes an open-source implementation of a phase-field model for brittle fracture using a recently developed finite-element toolbox, Gridap in Julia. This work exploits the advantages of both the phase-field model and Gridap toolbox for simulating fracture in brittle materials. On one hand, the use of the phase-field model, which is a continuum approach and uses a diffuse representation of sharp cracks, enables the proposed implementation to overcome such well-known drawbacks of the discrete approach for predicting complex crack paths as the need for re-meshing, enrichment of finite-element shape functions, and an explicit tracking of the crack surfaces. On the other hand, the use of Gridap makes the proposed implementation very compact and user-friendly that requires low memory usage, and provides a high degree of flexibility to the users in defining weak forms of partial differential equations. Tests on a single-edge notched plate under tension, an L-shaped panel, a notched plate with a hole, a notched beam under symmetric three-point bending and a notched beam with three holes under asymmetric three-point bending are considered to demonstrate how the proposed Gridap-based phase-field Julia code can be used to simulate fracture in brittle materials.

A rigorous and efficient explicit algorithm for irreversibility enforcement in phase-field finite element modeling of brittle crack propagation

In the present work, a computationally efficient and explicit algorithm for the rigorous enforcement of the irreversibility constraint in the phase-field modeling of brittle fracture is presented. The proposed approach is staggered and relies on the alternate minimization of the total energy functional. The phase-field evolution turns out to be governed by a complementarity boundary-value problem, where the complementarity stems from the irreversibility, while the boundary-value problem is originated by the presence of the gradient term in the phase-field functional. Several different techniques have been proposed in the literature to account for damage irreversibility in a computationally effective way. Following a similar approach proposed in the past for a gradient-plasticity problem, a particularly simple and effective solution strategy based on the Projected Successive Over-Relaxation (PSOR) method for constrained optimization, where an iterative explicit scheme is used for the solution of symmetric linear complementarity problems, is presented. Even though the proposed method is restricted to linear complementarity problems, it can be applied to the numerical simulation of a wide range of problems discussed in the literature. The performance of the suggested solution algorithm on a number of test cases is compared with that of a recently proposed penalty approach for the approximated enforcement of irreversibility.

Phase-field model for brittle fracture based on the inner-element edge-based smoothed finite method (IES-FEM)

In this work, a phase field fracture model based on the Inner-based Edge-based Smoothed Finite Element Method (IES-FEM) is implemented to simulate the brittle fracture in ABAQUS through subroutine UEL. The specific application of IES-FEM on phase field method (PFM) is explained. Through the single-edge notched tension/shear test and symmetric double notched tensile test, the differences of effects on numerical simulation of different length scale parameters, different mesh size and different displacement increment were discussed between IES-FEM-PFM and FEM-PFM. The coupled governing equations were solved alternately by decoupling the phase and displacement field. The results showed that the maximum reaction force of the load–displacement curve of IES-FEM-PFM is always smaller than that calculated by FEM-PFM, which was closer to the results of monolithic scheme. The stress and phase field variable at the same step time were also smaller than those of FEM-PFM, which is closer to the real stress field. After that some complex models were implemented to simulate the crack growth path in the phase field fracture model of IES-FEM. The results showed that IES-FEM did not change the characteristics of the phase field fracture model and the applicability and accuracy of IES-FEM in the phase field fracture model was verified effectively.

Nucleation under multi-axial loading in variational phase-field models of brittle fracture

Phase-field models of brittle fracture can be regarded as gradient damage models including an intrinsic internal length. This length determines the stability threshold of solutions with homogeneous damage and thus the strength of the material, and is often tuned to retrieve the experimental strength in uniaxial tensile tests. In this paper, we focus on multiaxial stress states and show that the available energy decompositions, introduced to avoid crack interpenetration and to allow for unsymmetric fracture behavior in tension and compression, lead to multiaxial strength surfaces of different but fixed shapes. Thus, once the length scale is tailored to recover the experimental tensile strength, it is not possible to match the experimental compressive or shear strength. We propose a new energy decomposition that enables the straightforward calibration of a multi-axial failure surface of the Drucker-Prager type. The new decomposition, which hinges upon the theory of structured deformations, encompasses the volumetric-deviatoric and the no-tension models as special cases. Preserving the variational structure of the model, it includes an additional free parameter that can be calibrated based on the experimental ratio of the compressive to the tensile strength (or, if possible, of the shear to the tensile strength), as successfully demonstrated on two data sets taken from the literature.

Adaptive analysis for phase-field model of brittle fracture of functionally graded materials

Gradient distributions of material properties are taken into account in phase-field model of fracture and the adaptive consistent element-free Galerkin method (EFG) for the fracture analysis of functionally graded materials (FGMs) is presented. Considering the fact that both the moving least-square approximation and the consistent integration scheme contribute to producing the accurate stress fields, the phase-field model of fracture for FGMs is numerically solved by the consistent EFG method in this paper. The corresponding adaptive criterion is established and 2D and 3D crack propagation of FGMs is simulated. Numerical results indicate that the proposed method can accurately reflect the influence of different distributions of material parameters on the crack propagation and has high efficiency compared with the standard EFG method and linear FEM. At some level, the fracture mechanism that crack propagation is driven by both the strain energy history and the critical energy release rate is revealed using phase-field model.

An accelerated staggered scheme for variational phase-field models of brittle fracture

There is currently an increasing interest in developing efficient solvers for variational phase-field models of brittle fracture. The governing equations for this problem originate from a constrained minimization of a non-convex energy functional, and the most commonly used solver is a staggered solution scheme. This is known to be robust compared to the monolithic Newton method, however, the staggered scheme often requires many iterations to converge when cracks are evolving. The focus of our work is to accelerate the solver through a scheme that sequentially applies Anderson acceleration and over-relaxation, switching back and forth depending on the residual evolution, and thereby ensuring a decreasing tendency. The resulting scheme takes advantage of the complementary strengths of Anderson acceleration and over-relaxation to make a robust and accelerating method for this problem. The new method is applied as a post-processing technique to the increments of the solver. Hence, the implementation merely requires minor modifications to already available software. Moreover, the cost of the acceleration scheme is negligible. The robustness and efficiency of the method are demonstrated through numerical examples.

Oscillatory and tip-splitting instabilities in 2D dynamic fracture: The roles of intrinsic material length and time scales

Recent theoretical and computational progress has led to unprecedented understanding of symmetry-breaking instabilities in 2D dynamic fracture. At the heart of this progress resides the identification of two intrinsic, near crack tip length scales — a nonlinear elastic length scale ℓ and a dissipation length scale ξ — that do not exist in Linear Elastic Fracture Mechanics (LEFM), the classical theory of cracks. In particular, it has been shown that at a propagation velocity v of about 90% of the shear wave-speed, cracks in 2D brittle materials undergo an oscillatory instability whose wavelength varies linearly with ℓ, and at larger loading levels (corresponding to yet higher propagation velocities), a tip-splitting instability emerges, both in agreements with experiments. In this paper, using phase-field models of brittle fracture, we demonstrate the following properties of the oscillatory instability: (i) It exists also in the absence of near-tip elastic nonlinearity, i.e. in the limit ℓ→0, with a wavelength determined by the dissipation length scale ξ. This result shows that the instability crucially depends on the existence of an intrinsic length scale associated with the breakdown of linear elasticity near crack tips, independently of whether the latter is related to nonlinear elasticity or to dissipation. (ii) It is a supercritical Hopf bifurcation, featuring a vanishing oscillations amplitude at onset. (iii) It is largely independent of the phenomenological forms of the degradation functions assumed in the phase-field framework to describe the cohesive zone, and of the velocity-dependence of the fracture energy Γ(v) that is controlled by the dissipation time scale in the Ginzburg–Landau-type evolution equation for the phase-field. These results substantiate the universal nature of the oscillatory instability in 2D. In addition, we provide evidence indicating that the tip-splitting instability is controlled by the limiting rate of elastic energy transport inside the crack tip region. The latter is sensitive to the wave-speed inside the dissipation zone, which can be systematically varied within the phase-field approach. Finally, we describe in detail the numerical implementation scheme of the employed phase-field fracture approach, allowing its application in a broad range of materials failure problems.

Comparative study of phase-field damage models for hydrogen assisted cracking

Hydrogen assisted cracking (HAC) usually causes premature failure of metallic materials and results in unexpected collapse of structures under the environmental exposure of hydrogen. Therefore, computational modeling of HAC is of paramount significance to quantify the adverse effects of HAC on the integrity and safety of structures. In this respect phase-field models for fracture are promising since they are able to seamlessly deal with complex crack patterns like nucleation, branching, merging and even fragmentation in a standalone framework. In this work we provide a comparative study of three frequently adopted phase-field models for hydrogen assisted cracking. Both the phase-field models for brittle fracture (e.g., the AT1/2 and WN models) and the phase-field regularized cohesive zone model (PF-CZM) are considered within the unified phase-field theory for damage and fracture, and are extended to incorporate the hydrogen enhanced decohesion mechanism. The numerical implementation of the phase-field models for HAC is also presented, with a simple scheme for calculating the gradient of the hydrostatic stress. Representative numerical examples show that, the PF-CZM with an increasing Irwin’s characteristic (internal) length and the PF-CZM with a constant one are both insensitive to the regularization length scale parameter. This merit make them promising for the computational modeling of HAC.

Phase-field model of brittle fracture in Reissner–Mindlin plates and shells

In this paper, a phase-field model is presented for the description of brittle fracture in a Reissner–Mindlin plate and shell formulation. The shell kinematics as well as the phase-field variable are described on the midsurface of the structure. Non-Uniform Rational B-Spline basis functions are used for the discretization of both the displacement/rotations and the phase-field. The spectral decomposition for the tension–compression split is applied on the total strain tensor, which varies through the thickness. Thus, the plane stress condition has to be enforced numerically. Various numerical examples are presented in order to verify the accuracy and effectiveness of the method and a detailed comparison to existing formulations is performed.

The use of Powell-Sabin B-Splines in a higher-order phase-field model for crack kinking

Phase-field models for brittle fracture in anisotropic materials result in a fourth-order partial differential equation for the damage evolution. This necessitates a $$\mathcal {C}^1$$ continuity of the basis functions. Here, Powell-Sabin B-splines, which are based on triangles, are used for the approximation of the field variables as well as for the the description of the geometry. The use of triangles makes adaptive mesh refinement and discrete crack insertion straightforward. Bezier extraction is used to cast the B-splines in a standard finite element format. A procedure to impose Dirichlet boundary condition is provided for these elements. The versatility and accuracy of the approach are assessed in two case studies, featuring crack kinking and zig-zag crack propagation. It is also shown that the adaptive refinement well captures the evolution of the phase field.

Stochastic phase-field modeling of brittle fracture: Computing multiple crack patterns and their probabilities

In variational phase-field modeling of brittle fracture, the functional to be minimized is not convex, so that the necessary stationarity conditions of the functional may admit multiple solutions. The solution obtained in an actual computation is typically one out of several local minimizers. Evidence of multiple solutions induced by small perturbations of numerical or physical parameters was occasionally recorded but not explicitly investigated in the literature. In this work, we focus on this issue and advocate a paradigm shift, away from the search for one particular solution towards the simultaneous description of all possible solutions (local minimizers), along with the probabilities of their occurrence. Inspired by recent approaches advocating measure-valued solutions (Young measures as well as their generalization to statistical solutions) and their numerical approximations in fluid mechanics, we propose the stochastic relaxation of the variational brittle fracture problem through random perturbations of the functional. We introduce the concept of stochastic solution, with the main advantage that point-to-point correlations of the crack phase fields in the underlying domain can be captured. These stochastic solutions are represented by random fields or random variables with values in the classical deterministic solution spaces. In the numerical experiments, we use a simple Monte Carlo approach to compute approximations to such stochastic solutions. The final result of the computation is not a single crack pattern, but rather several possible crack patterns and their probabilities. The stochastic solution framework using evolving random fields allows additionally the interesting possibility of conditioning the probabilities of further crack paths on intermediate crack patterns.

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.

Using Deep Learning to Predict Fracture Patterns in Crystalline Solids

Summary Fracture is a catastrophic process whose understanding is critical for evaluating the integrity and sustainability of engineering materials. Here, we present a machine-learning approach to predict fracture processes connecting molecular simulation into a physics-based data-driven multiscale model. Based on atomistic modeling and a novel image-processing approach, we compile a comprehensive training dataset featuring fracture patterns and toughness values for different crystal orientations. Assessments of the predictive power of the machine-learning model shows excellent agreement not only regarding the computed fracture patterns but also the fracture toughness values and is examined for both mode I and mode II loading conditions. We further examine the ability of predicting fracture patterns in bicrystalline materials and material with gradients of microstructural crystal orientation. These results further underscore the excellent predictive power of our model. Potential applications of this model could be widely applied in material design.

An auto-adaptive sub-stepping algorithm for phase-field modeling of brittle fracture

In this work, we propose an auto-adaptive displacement stepping algorithm to auto-adaptively adjust the displacement step size while solving the quasi-static brittle fracture propagation problem with the phase-field method. We employ a step size controller based on an ‘a posteriori’ error estimator to achieve adaptivity in the displacement step size. To capture correct crack propagation behavior near peak load, we propose an energy-based conditional. The conditional is evaluated in a sub-step of the primary displacement step, and it decides whether to accept or reject the step. Based on the step size controller algorithm and the sub-stepping algorithm, we formulate the auto-adaptive sub-stepping algorithm. We investigate the accuracy and computational cost of the proposed algorithm on several benchmark problems from the literature and compare it to the standard alternate minimization (AM) approach. The algorithm auto-adaptively adjusts the size of the displacement step and allows us to use step sizes that may vary by even three orders of magnitude. As compared to the standard AM algorithm, this results in a reduction of 78–90% in the CPU time required to reach peak reaction force, and 56–78% reduction in the CPU time needed to complete analysis, and always maintains the accuracy in crack path prediction.

Adaptive refinement for phase-field models of brittle fracture based on Nitsche’s method

A new adaptive refinement strategy for phase-field models of brittle fracture is proposed. The approach provides a computationally efficient solution to the high demand in spatial resolution of phase-field models. The strategy is based on considering two types of elements: h-refined elements along cracks, where more accuracy is needed to capture the solution, and standard elements in the rest of the domain. Continuity between adjacent elements of different type is imposed in weak form by means of Nitsche’s method. The weakly imposition of continuity leads to a very local refinement in a simple way, for any degree of approximation and both in 2D and 3D. The performance of the strategy is assessed for several scenarios in the quasi-static regime, including coalescence and branching of cracks in 2D and a twisting crack in 3D.

Stabilized mixed formulation for phase-field computation of deviatoric fracture in elastic and poroelastic materials

In the numerical approximation of phase-field models of fracture in porous media with the finite element method, the problem of numerical locking may occur. The causes can be traced both to the hydraulic and to the mechanical properties of the material. In this work we present a mixed finite element formulation for phase-field modeling of brittle fracture in elastic solids based on a volumetric-deviatoric energy split and its extension to water saturated porous media. For the latter, two alternative mixed formulations are proposed. To be able to use finite elements with linear interpolation for all the field variables, which violates the Ladyzenskaja–Babuska–Brezzi condition, a stabilization technique based on polynomial pressure projections, proposed and tested by previous authors in fluid mechanics and poromechanics, is introduced. We develop an extension of this stabilization to phase-field mixed models of brittle fracture in porous media. Several numerical examples are illustrated, to show the occurrence of different locking phenomena and to compare the solutions obtained with different unstable, stable and stabilized low order finite elements.

A recursive multilevel trust region method with application to fully monolithic phase-field models of brittle fracture

The simulation of crack initiation and propagation in an elastic material is difficult, as crack paths with complex topologies have to be resolved. Phase-field approaches allow to simulate crack behavior without the need to explicitly model crack paths. However, the underlying mathematical model gives rise to a non-convex constrained minimization problem. In this work, we propose a recursive multilevel trust region (RMTR) method to efficiently solve such a minimization problem. The RMTR method combines global convergence properties of trust region methods with the optimality of multilevel methods. The solution process is accelerated by employing level dependent objective functions, minimization of which provides correction to the original/fine-level problem. In the context of the phase-field fracture approach, it is challenging to design efficient level dependent objective functions as a certain mesh resolution is required to resolve fracture bands. We introduce level dependent objective functions that combine fine level description of the crack path with the coarse level discretization. The overall performance and the convergence properties of the proposed RMTR method are investigated by means of several numerical examples in three dimensions.