Phase-field Modeling Of Brittle Fracture Research Articles

A variational phase-field model for brittle fracture in polydisperse elastomer networks

The distribution of chain lengths in elastomer networks occurs naturally during polymerization, and has a critical role in the mechanical behavior of these materials, including the damage and fracture response. A common underlying assumption for the majority of constitutive models for rubber elasticity is that all the chains admit the same length. Moreover, in the classical statistical mechanical model for elastomer networks, the changes in internal energy are generally assumed to be negligible in comparison to the changes in configurational entropy. In contrast, the fracture process in a elastomer network, as already demonstrated in the well-known Lake-Thomas model, is essentially internal energy dominated. In this paper, we formulate a phase-field model for brittle fracture in polydisperse elastomer networks extending and merging recent advances in the fields of (a) homogenization of elastomer networks and (b) the variational approach to brittle fracture, allowing for predictions of crack nucleation, initiation and propagation. The free energy of the continuum is obtained employing an eight chain network model, accounting for (a) internal energy contributions from the extension of molecular bonds and (b) arbitrary chain length distribution. The representative chain in the eight chain network model takes into account the distribution of chain lengths, following the recently developed equal force model. We employ a mixed displacement-pressure formulation for the discretization of the incompressible large deformation elastic problem that arises. The analytical solution for the crack nucleation problem of an incompressible hyperelastic bar under uniaxial loading is compared with the three-dimensional simulation result. Finally, we demonstrate through a representative numerical simulation the capability of the gradient damage model to simulate crack propagation in an incompressible elastomer network.

A phase-field model for fatigue crack growth

A phase-field model is presented for the study of fatigue crack growth. A crack growth viscosity parameter is introduced into the standard phase-field model for brittle fracture to account for rate- or cycle-dependent crack growth phenomena. A modified J-integral is developed to demonstrate how the phase-field approach can be used to generate Paris-law type crack growth rates. In order to model more general crack growth versus applied loading behaviors that are not a single simple Paris-law, steady-state finite element calculations are performed to calibrate fits of the phase-field model to measured crack growth rates found in da/dN versus ΔK curves. Transient time- or cycle-dependent calculations are performed and compared to experimental measurements on samples where crack turning is induced by the presence of a hole in the vicinity of the crack. A three-dimensional example with turning of the crack front is also computed illustrating the capabilities of the phase-field approach for complex crack paths.

Adaptive phase-field modeling of brittle fracture using the scaled boundary finite element method

In this work, we propose an adaptive phase field method (PFM) to simulate quasi-static brittle fracture problems. The phase field equations are solved using the scaled boundary finite element method (SBFEM). The adaptive refinement strategy is based on an error indicator evaluated directly from the solutions of the SBFEM without any need for stress recovery techniques. Quadtree meshes are adapted to perform mesh refinement. The polygons with hanging nodes in the quadtree decomposition are treated as n−sided polygons within the framework of the SBFEM and do not require any special treatment in contrast to the conventional finite element method. Several benchmark problems are used to demonstrate the robustness and the efficacy of the proposed technique. The adaptive refinement strategy reduces the mesh burden when adopting the PFM to model fracture. Numerical results show an improvement in the computational efficiency in terms of the number of elements required in the standard PFM without compromising the accuracy of the solution.

Mesh-Based and Meshfree Reduced Order Phase-Field Models for Brittle Fracture: One Dimensional Problems.

Modelling brittle fracture by a phase-field fracture formulation has now been widely accepted. However, the full-order phase-field fracture model implemented using finite elements results in a nonlinear coupled system for which simulations are very computationally demanding, particularly for parametrized problems when the randomness and uncertainty of material properties are considered. To tackle this issue, we present two reduced-order phase-field models for parametrized brittle fracture problems in this work. The first one is a mesh-based Proper Orthogonal Decomposition (POD) method. Both the Discrete Empirical Interpolation Method (DEIM) and the Matrix Discrete Empirical Interpolation Method ((M)DEIM) are adopted to approximate the nonlinear vectors and matrices. The second one is a meshfree Krigingmodel. For one-dimensional problems, served as proof-of-concept demonstrations, in which Young’s modulus and the fracture energy vary, the POD-based model can speed up the online computations eight-times, and for the Kriging model, the speed-up factor is 1100, albeit with a slightly lower accuracy. Another merit of the Kriging’s model is its non-intrusive nature, as one does not need to modify the full-order model code.

On penalization in variational phase-field models of brittle fracture

Irreversible evolution is one of the central concepts as well as implementation challenges of both the variational approach to fracture by Francfort and Marigo (1998) and its regularized counterpart by Bourdin, Francfort and Marigo (2000, 2007 and 2008), which is commonly referred to as a phase-field model of brittle fracture. Irreversibility of the crack phase-field imposed to prevent fracture healing leads to a constrained minimization problem, whose optimality condition is given by a variational inequality. In our study, the irreversibility is handled via penalization. Provided the penalty constant is well-tuned, the penalized formulation is a good approximation to the original one, with the advantage that the induced equality-based weak problem enables a much simpler algorithmic treatment. We propose an analytical procedure for deriving the optimal penalty constant, more precisely, its lower bound, which guarantees a sufficiently accurate enforcement of the crack phase-field irreversibility. Our main tool is the notion of the optimal phase-field profile, as well as the Γ-convergence result. It is shown that the explicit lower bound is a function of two formulation parameters (the fracture toughness and the regularization length scale) but is independent of the problem setup (geometry, boundary conditions etc.), the formulation ingredients (degradation function, tension–compression split etc.), as well as the discretization (mesh size). The optimally-penalized formulation is tested for three benchmark problems, including one with available analytical solution. We also compare our results with those obtained by using the alternative irreversibility technique based on the notion of the history field by Miehe et al. (2010).

Adaptive consistent element-free Galerkin method for phase-field model of brittle fracture

Efficient implementation of the element-free Galerkin (EFG) method for a phase-field model of linear elastic fracture mechanics is presented, in which the convenience of the meshfree method to construct high order approximation functions and to implement h-adaptivity is fully exploited. A second-order moving-least squares approximation for both displacement and phase field is employed. Domain integration of the weak forms is evaluated by the quadratically consistent 3-point integration scheme. The refinement criterion using maximum residual strain energy history is proposed and the insertion of nodes is based on the background mesh. Numerical results show that the developed method is more efficient than the standard finite element method (3-node triangle element) due to the proposed h-adaptivity. In comparison with the standard EFG method, the proposed consistent EFG method significantly improves the computational efficiency and accuracy. The advantage of the quadratic approximation is also demonstrated. In addition, the feasibility of extending the proposed method to 3D is validated by the modeling of a twisting crack.

A FFT solver for variational phase-field modeling of brittle fracture

The variational phase-field method is an attractive non-local approach of modeling fracture in heterogeneous materials. However, these materials usually require a fine mesh to resolve the fracture process zone. Consequently, the standard finite element solver becomes cumbersome due to the large number of elements in applications with highly heterogeneous materials. Motivated by this limitation, an algorithm based on FFT methods has been introduced in this paper to solve the phase-field model of brittle fracture. Relying on a staggered update scheme, the proposed algorithm solves the fracture problem and mechanical problem separately, both using the FFT technique. It inherits the advantages of classical FFT methods in terms of simplicity of mesh generation and parallel implementation. Introduced within a FFT-based code “AMITEX”, it takes the advantage of massively parallel capabilities associated with a distributed memory implementation. The characteristics of the proposed method are analyzed in a single edge notched specimen benchmark. Representative numerical examples demonstrate that the proposed FFT solver is capable of predicting different crack modes and complex crack configuration, such as crack interaction, branching and coalescence. Finally, a model of an idealized continuous fiber composite with void involving over 32 million voxels is solved, illustrating the potential of the FFT solver in large-scale problems.

Crack kinking in a variational phase-field model of brittle fracture with strongly anisotropic surface energy

In strongly anisotropic materials the orientation-dependent fracture surface energy is a non-convex function of the crack angle. In this context, the classical Griffith model becomes ill-posed and requires a regularization. We revisit the crack kinking problem in materials with strongly anisotropic surface energies by using a variational phase-field model. The model includes in the energy functional a quadratic term on the second gradient of the phase-field. This term has a regularizing effect, energetically penalizing the crack curvature. We provide analytical formulas for the dependence of the surface energy on the crack direction and develop an open-source finite-element solver for the higher-order phase-field problem. Quantitative numerical experiments for the crack kinking problem show that the crack kinking directions observed in our phase-field simulations are in close agreement with the generalized maximum energy release rate criterion. Finally, we revisit a thermal quenching experiment in the case of slabs with strongly anisotropic surface energies. We show that the anisotropy can strongly affect the observed crack patterns, either by stabilizing straight cracks or by inducing zig-zag crack patterns. In the case of zig-zag cracks, we observe that crack kinking is always associated with an unstable propagation of a finite length add-crack in a single time-step.

Consistent finite-dimensional approximation of phase-field models of fracture

In this paper we focus on the finite-dimensional approximation of quasi-static evolutions of critical points of the phase-field model of brittle fracture. In a space discretized setting, we first discuss an alternating minimization scheme which, together with the usual time-discretization procedure, allows us to construct such finite-dimensional evolutions. Then, passing to the limit as the space discretization becomes finer and finer, we prove that any limit of a sequence of finite-dimensional evolutions is itself a quasi-static evolution of the phase-field model of fracture. In particular, our proof shows for the first time the consistency of numerical schemes related to the study of fracture mechanics and image processing.

A phase-field model for strength and fracture analyses of fiber-reinforced composites

A phase-field model for brittle fracture is proposed and evaluated for strength and fracture analyses of composites. In addition to the elastic properties, this approach makes use of only the fracture toughness and the strength of the material. The capability of the method is shown in analyses of composites at two scales. In laminates, strengths of notched laminates are estimated, including hole size effects. In a lamina, cracks developed in both transverse tension and compression are analyzed and compared to other numerical methods in the literature. The effects of a spectral and a hydrostatic-deviatoric decomposition of the strain energy density, two variants often used in phase-field formulations, are studied. It is shown that the choice of the decomposition affects the fracture development. Results are compared to experiments and simulations in the literature showing the capabilities of the phase-field approach.

Phase-field modeling of brittle fracture with multi-level hp-FEM and the finite cell method

The difficulties in dealing with discontinuities related to a sharp crack are overcome in the phase-field approach for fracture by modeling the crack as a diffusive object being described by a continuous field having high gradients. The discrete crack limit case is approached for a small length-scale parameter that controls the width of the transition region between the fully broken and the undamaged phases. From a computational standpoint, this necessitates fine meshes, at least locally, in order to accurately resolve the phase-field profile. In the classical approach, phase-field models are computed on a fixed mesh that is a priori refined in the areas where the crack is expected to propagate. This on the other hand curbs the convenience of using phase-field models for unknown crack paths and its ability to handle complex crack propagation patterns. In this work, we overcome this issue by employing the multi-level hp-refinement technique that enables a dynamically changing mesh which in turn allows the refinement to remain local at singularities and high gradients without problems of hanging nodes. Yet, in case of complex geometries, mesh generation and in particular local refinement becomes non-trivial. We address this issue by integrating a two-dimensional phase-field framework for brittle fracture with the finite cell method (FCM). The FCM based on high-order finite elements is a non-geometry-conforming discretization technique wherein the physical domain is embedded into a larger fictitious domain of simple geometry that can be easily discretized. This facilitates mesh generation for complex geometries and supports local refinement. Numerical examples including a comparison to a validation experiment illustrate the applicability of the multi-level hp-refinement and the FCM in the context of phase-field simulations.

A residual control staggered solution scheme for the phase-field modeling of brittle fracture

The phase-field approach to fracture modeling circumvents the crack-surface tracking problem by introducing the phase-field variable which separates the broken and unbroken material states through a smooth transition. However, very fine spatial discretization is required to resolve the smooth distribution of the phase-field regulated by a small length scale parameter. Thus, it can be computationally rather expensive when paired with an inefficient solution scheme. This contribution presents a comprehensive discussion on the use of a stopping criterion within the broadly used staggered algorithm. The stopping criterion of the iterative scheme based on the control of the residual norm is introduced and implemented in the finite element software Abaqus. It alleviates the problem of the common single iteration staggered algorithm, which requires fine loading incrementation to produce an accurate solution. The model verification is conducted on several standard benchmark tests and a porous microstructure exhibiting complex crack phenomena. The detailed discussions regarding the proposed implementation’s accuracy and CPU time usage are provided, demonstrating an improvement in computational efficiency with accurate results no longer dependent on the careful selection of loading incrementation. The algorithm codes in the form of the Abaqus user subroutines UMAT and UEL are publicly available on Mendeley data repository linked to this work.

Phase-field modeling of brittle fracture using an efficient virtual element scheme

This work addresses an efficient low order virtual element method (VEM) for the phase-field modeling of isotropic brittle fracture. Virtual elements were introduced in the last decade and applied to various problems in solid mechanics. The phase-field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with constitutive terms rooted in fracture mechanics, see Miehe et al. [23] and Miehe et al. [29]. In the presented contribution, we propose a rigorous variational-based framework for the phase-field modeling of brittle fracture in elastic solids undergoing small strains. The key goal here, is the extension towards the recently developed virtual element formulation due to the flexible choice of nodes number in an element which can be changed easily during the simulation process, as outlined in Wriggers et al. [11] and Wriggers et al. [18]. To this end, the potential density is formulated in terms of suitable polynomial functions, instead of computing the unknown shape functions for complicated element geometries, e.g. arbitrary convex or concave polygonal elements. An important aspect of this work is the introduction of an incremental minimization principle, with a novel construction of the stabilization density for the coupled multi-field problem. On the computational side, a robust and efficient monolithic scheme is employed using the software tool AceFEM program in the numerical implementation to compute the unknowns (displacement and crack phase-field), see Korelc and Wriggers [52]. The performance of the formulation is underlined by means of representative examples.

A length scale insensitive phase-field damage model for brittle fracture

Being able to model complex nucleation, propagation, branching and merging of cracks in solids within a unified framework, the classical phase-field models for brittle fracture fail in predicting length scale independent global responses for a solid lacking elastic singularities (e.g., corners, notches, etc.). Motivated from Barenblatt’s approximation of Griffith’s brittle fracture with a vanishing Irwin’s internal length, this paper extends our recent work in quasi-brittle failure (Wu, 2017, 2018a) and presents for the first time a length scale insensitive phase-field damage model for brittle fracture. More specifically, with a set of optimal characteristic functions, a phase-field regularized cohesive zone model (CZM) with linear softening law is addressed and applied to brittle fracture. Both the failure strength and the traction – separation law are independent of the incorporated length scale parameter. Compared to other phase-field models and CZM based discontinuous approaches for brittle fracture, the proposed phase-field regularized CZM is of several merits. On the one hand, being theoretically equivalent to Barenblatt’s CZM (at least in the 1-D case), it needs neither the explicit crack representation/tracking nor the elastic penalty stiffness which both are necessary but cumbersome for discontinuous approaches. On the other hand, it gives length scale independent global responses for problems with or without elastic singularities while preserving the expected Γ-convergence property of phase-field models. Representative numerical examples of several well-known benchmark tests support the above conclusions, validating its capability of modeling both mode-I and mixed-mode brittle fracture.

A non-intrusive global/local approach applied to phase-field modeling of brittle fracture

This paper aims at investigating the adoption of non-intrusive global/local approaches while modeling fracture by means of the phase-field framework. A successful extension of the non-intrusive global/local approach to this setting would pave the way for a wide adoption of phase-field modeling of fracture, already well established in the research community, within legacy codes for industrial applications. Due to the extreme difference in stiffness between the global counterpart of the zone to be analized locally and its actual response when undergoing extensive cracking, the main foreseen issues are robustness, accuracy and efficiency of the fixed point iterative algorithm which is at the core of the method. These issues are tackled in this paper. We investigate the convergence performance when using the native global/local algorithm and show that the obtained results are identical to the reference phase-field solution. We also equip the global/local solution update procedure with relaxation/acceleration techniques such as Aitken’s Delta ^2-method, the Symmetric Rank One and Broyden’s methods and show that the iterative convergence can be improved significantly. Results indicate that Aitken’s Delta ^2-method is probably the most convenient choice for the implementation of the approach within legacy codes, as this method needs only tools already available for the so-called sub-modeling approach, a strategy routinely used in industrial contexts.

Moving mesh finite element simulation for phase-field modeling of brittle fracture and convergence of Newton's iteration

A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fracture. The moving mesh partial differential equation approach is employed to dynamically track crack propagation. Meanwhile, the decomposition of the strain tensor into tensile and compressive components is essential for the success of the phase-field modeling of brittle fracture but results in a non-smooth elastic energy and stronger nonlinearity in the governing equation. This makes the governing equation much more difficult to solve and, in particular, Newton's iteration often fails to converge. Three regularization methods are proposed to smooth out the decomposition of the strain tensor. Numerical examples of fracture propagation under quasi-static load demonstrate that all of the methods can effectively improve the convergence of Newton's iteration for relatively small values of the regularization parameter but without compromising the accuracy of the numerical solution. They also show that the moving mesh finite element method is able to adaptively concentrate the mesh elements around propagating cracks and handle multiple and complex crack systems.

High-accuracy phase-field models for brittle fracture based on a new family of degradation functions

Phase-field approaches to fracture based on energy minimization principles have been rapidly gaining popularity in recent years, and are particularly well-suited for simulating crack initiation and growth in complex fracture networks. In the phase-field framework, the surface energy associated with crack formation is calculated by evaluating a functional defined in terms of a scalar order parameter and its gradients. These in turn describe the fractures in a diffuse sense following a prescribed regularization length scale. Imposing stationarity of the total energy leads to a coupled system of partial differential equations that enforce stress equilibrium and govern phase-field evolution. These equations are coupled through an energy degradation function that models the loss of stiffness in the bulk material as it undergoes damage. In the present work, we introduce a new parametric family of degradation functions aimed at increasing the accuracy of phase-field models in predicting critical loads associated with crack nucleation as well as the propagation of existing fractures. An additional goal is the preservation of linear elastic response in the bulk material prior to fracture. Through the analysis of several numerical examples, we demonstrate the superiority of the proposed family of functions to the classical quadratic degradation function that is used most often in the literature.

Hybrid nonlinear surrogate models for fracture behavior of polymeric nanocomposites

We present a hybrid nonlinear surrogate model for fracture in polymeric nanocomposites. The phase field method is employed to model fracture in the polymer matrix. Since the stochastic analysis on the output of the mechanical model is prohibitively expensive, surrogate models (SM) are very attractive alternatives. In order to get an optimal and robust solution, we propose a hybrid nonlinear surrogate model (HSM) for the prediction of the fracture toughness of PNC. It is constructed with the use of the polynomial regression and the Kriging interpolation. The support data for such HSM is generated by a phase-field model for brittle fracture with six chosen input parameters. The validation of the surrogate model and by this its qualitative assessment is done based on a scanning test set algorithm. The constructed and assessed HSM is then used to present the behavior of fracture toughness of PNC with respect to various input parameters with very low computational costs and high accuracy. Within the domain of interest, the analysis shows that Young’s modulus of the matrix has no optimum value, in which, the higher input value causes higher response. On the other hand the volume fraction of clay platelets at about 5% showed stability of the response, in which, the higher input value leads to no change in the response.

A convergence study of phase-field models for brittle fracture

A crucial issue in phase-field models for brittle fracture is whether the functional that describes the distributed crack converges to the functional of the discrete crack when the internal length scale introduced in the distribution function goes to zero. Theoretical proofs exist for the original theory. However, for continuous media as well as for discretised media, significant errors have been reported in numerical solutions regarding the approximated crack surface, and hence for the dissipated energy. We show that for a practical setting, where the internal length scale and the spacing of the discretisation are small but finite, the observed discrepancy partially stems from the fact that numerical studies consider specimens of a finite length, and partially relates to the irreversibility introduced when casting the variational theory for brittle fracture in a damage-like format. While some form of irreversibility may be required in numerical implementations, the precise form significantly influences the accuracy and convergence towards the discrete crack.

Crack nucleation in variational phase-field models of brittle fracture

Phase-field models, sometimes referred to as gradient damage or smeared crack models, 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’ criterion. For a one-dimensional problem, it has been shown that they can predict nucleation upon a critical stress, provided that the regularization parameter be identified with the material’s internal or characteristic length. In this article, 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 that predicted by a strength criterion to that of a toughness criterion when the strength of the stress concentration or singularity varies. We present validation and verification 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.Our main claim, supported by validation and verification in a broad range of materials and geometries, is that crack nucleation can be accurately predicted by minimization of a nonlinear energy in variational phase field models, and does not require the introduction of ad-hoc criteria.