Phase-field Modeling Of Brittle Fracture Research Articles

Calibration and validation of a phase-field model of brittle fracture within the damage mechanics challenge

In the context of the Damage Mechanics Challenge, we adopt a phase-field model of brittle fracture to blindly predict the behavior up to failure of a notched three-point-bending specimen loaded under mixed-mode conditions. The beam is additively manufactured using a geo-architected gypsum based on the combination of bassanite and a water-based binder. The calibration of the material parameters involved in the model is based on a set of available independent experimental tests and on a two-stage procedure. In the first stage an estimate of most of the elastic parameters is obtained, whereas the remaining parameters are optimized in the second stage to minimize the discrepancy between the numerical predictions and a set of experimental results on notched three-point-bending beams. The good agreement between numerical predictions and experimental results in terms of load–displacement curves and crack paths demonstrates the predictive ability of the model and the reliability of the calibration procedure.

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.

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.

A computational meshfree RPIM approach for phase-field modeling of brittle fracture

A computational meshfree RPIM approach for phase-field modeling of brittle fracture

Phase-field modeling of anisotropic crack propagation based on higher-order nonlocal operator theory

This paper presents a novel higher-order nonlocal operator theory for the phase-field modeling of brittle fracture in anisotropic materials. Incorporating higher order nonlocal operators can enhance the accuracy of the phase-field model by effectively capturing long-range interactions that hold significance in numerous materials. The reproducing kernel particle method is employed to derive a nonlocal differential operator to enhance computational stability and accuracy. Moreover, the proposed method eliminates the need for direct computation of derivatives of the modified kernel function, which avoids the calculation of moment matrix derivatives and improves computational efficiency. The phase-field modeling of polycrystalline materials, considering the anisotropic fracture resistance of each grain, is implemented using this numerical framework. The present method is able to capture different scenarios intergranular and transgranular crack propagation patterns in polycrystalline materials. The proposed method involves a detailed representation of the complex process of crack initiation and propagation in 2D and 3D models of polycrystalline materials.

Two different micro-polar phase-field models for brittle fracture and their open-source finite element implementation

Realizing the limitations of classical phase-field fracture models and the need for the development of non-local phase-field models that can capture the experimentally observed size effects in brittle and quasi-brittle materials, we propose thermodynamically consistent volumetric-deviatoric and spectral decomposition-based phase-field models (PFMs) for brittle fracture in micro-polar continua. For developing the proposed models, we have considered the micro-rotation as an additional kinematic descriptor, and have derived the balance equations by invoking the virtual power principle. On satisfying the thermodynamic laws, we have determined the constitutive relations for the thermodynamic fluxes and demonstrated that any dissipative effect can easily be incorporated into the proposed models. We have provided an open-source finite element (FE) implementation of the micro-polar PFMs using Gridap in Julia. The implementation includes various benchmark problems to illustrate how the responses in a specimen change with the variations in micro-polar material parameters. Through a series of numerical examples with different values of micro-polar material parameters, we have showcased both the capabilities and limitations of the proposed volumetric-deviatoric and spectral decomposition-based micro-polar PFMs.

A finite-volume implementation of the phase-field model for brittle fracture with adaptive mesh refinement

The phase-field method (PFM) has advantages in modeling crack propagation in rock-like materials by treating the crack surface as a continuous function, therefore avoiding dealing with the sharp discontinuous displacement field. This study presents an implementation of PFM using the finite volume method (FVM) to discretize the governing equations for the conservation of linear momentum and the evolution of phase field. The coupled equations are solved by an iterative staggered scheme. The adaptive mesh refinement (AMR) technique is further employed to improve computational efficiency, which is relatively easy to achieve in FVM as the method can naturally handle unstructured meshes with hanging nodes in both two- and three-dimensional problems, as opposed to the finite element method (FEM). Several classical crack propagation problems, including the single-edge notched tension and shear tests, the L-shaped panel test, and the test on a notched plate with a hole, are simulated and compared with available experimental and FEM results, which demonstrate that the FVM-based PFM can model crack propagation accurately and effectively and could be more efficient than the FEM-based PFM.

Representing model uncertainties in brittle fracture simulations

This work focuses on the representation of model-form uncertainties in phase-field models of brittle fracture. Such uncertainties can arise from the choice of the degradation function for instance, and their consideration has been unaddressed to date. The stochastic modeling framework leverages recent developments related to the analysis of nonlinear dynamical systems and relies on the construction of a stochastic reduced-order model. In the latter, a POD-based reduced-order basis is randomized using Riemannian projection and retraction operators, as well as an information-theoretic formulation enabling proper concentration in the convex hull defined by a set of model proposals. The model thus obtained is mathematically admissible in the almost sure sense and involves a low-dimensional hyperparameter, the calibration of which is facilitated through the formulation of a quadratic programming problem. The relevance of the modeling approach is further assessed on one- and two-dimensional applications. It is shown that model uncertainties can be efficiently captured and propagated to macroscopic quantities of interest. An extension based on localized randomization is also proposed to handle the case where the forward simulation is highly sensitive to sample localization. This work constitutes a methodological development allowing phase-field predictions to be endowed with statistical measures of confidence, accounting for the variability induced by modeling choices.

Phase-field models of floe fracture in sea ice

Abstract. We develop a phase-field model of brittle fracture to model fracture in sea ice floes. Phase fields allow for a variational formulation of fracture by using an energy functional that combines a linear elastic energy with a term modeling the energetic cost of fracture. We study the fracture strength of ice floes with stochastic thickness variations under boundary forcings or displacements. Our approach models refrozen cracks or other linear ice impurities with stochastic models for thickness profiles. We find that the orientation of thickness variations is an important factor for the strength of ice floes, and we study the distribution of critical stresses leading to fracture. Potential applications to discrete element method (DEM) simulations and field data from the ICEX 2018 campaign are discussed.

Phase-field modeling of brittle fracture in functionally graded materials using exponential finite elements

This study introduces a novel implementation of exponential finite element (EFE) shape functions within the phase field model for predicting fracture responses in functionally graded materials (FGMs). The proposed approach utilizes an effective fracture toughness concept to analyze load–displacement responses and crack paths in various examples of FGMs. To optimize computational efficiency, a mixed scheme combining both linear finite element (LFE) and EFE shape functions is employed. Specifically, only the critical elements are corrected using EFE shape functions. Comparative analysis of simulation results against converged outcomes demonstrates the superiority of the EFE scheme, even when employing coarser meshes. The EFE approach accurately predicts load–displacement responses and crack paths for the evaluated examples. The proposed implementation offers a reliable and efficient tool for studying fracture behavior in FGMs. Despite the higher integration schemes and orientation requirements associated with EFE shape functions, the additional computational effort is found to be negligible. Implementation aspects include a staggered iteration scheme, a hybrid tension–compression splitting scheme, and automatic orientation of EFE shape functions.

A phase-field model for thermo-elastic fracture in quasicrystals

In this paper, a thermo-elastic phase-field model for brittle fracture in quasicrystals is developed. The kind of quasicrystals is not preset, so this model is suitable for almost all thermo-elastic quasicrystal solids. The phonon force and thermal load trigger the variation of the phonon elastic energy, and then drive crack initiation and propagation. The thermal conductivity in the cracked region is assumed to be degraded, which means cracks are adiabatic. For the static penny-shaped crack problem, the results based on the present model agree well with the existing analytical solution. For the quasi-static crack problems, both the tensile/shear fracture in a thermal environment and the thermal shock cracking failure can be dealt with by the present model. Several examples of 1D hexagonal and 2D decagonal quasicrystals are performed to evaluate the effects of thermal load and phason elastic field. The simulation results show that the thermal load can influence the peak force and fracture displacement. For the single-edge notched tension test, the effect of the thermal load on the peak force is small, but it on the fracture displacement is obvious. For the single-edge notched shear test, the thermal load has an obvious influence on both the peak force and fracture displacement. For the thermal shock cracking problem, the phason elastic field has a significant effect on the cracking time and crack number. The present model can serve as a powerful tool to simulate various thermo-elastic fracture problems in quasicrystals.

A concise review of small-strain phase-field modeling of ductile fracture

The computational modeling of ductile fracture is a complex problem due to the simultaneous presence of various competing dissipation mechanisms at the microscale resulting in evolving displacement discontinuities at the macroscale. Phase-field modeling of brittle fracture has emerged as one of the more computationally efficient smeared approaches for simulating crack propagation in solids. In recent years, there has been significant attention towards extending this approach to ductile fracture. The first papers on this topic were published approximately a decade ago, and since then, there has been a growing number of new contributions proposing different approaches and applications. The purpose of this work is to offer an extensive bibliography on phase-field modeling of ductile fracture, together with a concise presentation of some of the most widely used formulations, specifically limited to small-strain plasticity under monotonic loading. Few aspects of particular interest are also discussed in detail.

An SPH-based FSI framework for phase-field modeling of brittle fracture under extreme hydrodynamic events

An SPH-based FSI framework for phase-field modeling of brittle fracture under extreme hydrodynamic events

A coupled approach to predict cone-cracks in spherical indentation tests with smooth or rough indenters

Indentation tests are largely exploited in experiments to characterize the mechanical and fracture properties of the materials from the resulting crack patterns. This work proposes an efficient theoretical and computational framework, whose implementation is detailed for 2D axisymmetric and for 3D geometries, to simulate indentation-induced cracking phenomena caused by non-conforming contacts with indenter profiles of an arbitrary shape. The formulation hinges on the coupling of the MPJR (eMbedded Profile for Joint Roughness) interface finite elements which embed the indenter profile to efficiently solve the contact problem between non-planar bodies, and the phase-field model for brittle fracture to simulate crack evolution and nonlocal damage in the substrate. The novel framework is applied to predict cone-crack formation in the case of indentation tests with smooth spherical indenters, with validation against experimental data. Then, the methodology is employed for the very first time in the literature to assess the effect of surface roughness superimposed on the shape of the smooth spherical indenter. In terms of physical insights, numerical predictions quantify the dependencies of the critical load for crack nucleation and the crack radius on the amplitude of roughness in comparison with the behavior of smooth indenters. Again, the consistency with available experimental trends is noticed.

Fast staggered schemes for the phase-field model of brittle fracture based on the fixed-stress concept

Phase-field models are promising to tackle various fracture problems where a diffusive crack is introduced and modeled using the phase variable. Owing to the non-convexity of the energy functional, the derived partial differential equations are usually solved in a staggered manner. However, this method suffers from a low convergence rate, and a large number of staggered iterations are needed, especially at the fracture nucleation and propagation. In this study, we propose novel staggered schemes, which are inspired by the fixed-stress split scheme in poromechanics. By fixing certain invariants of the stress when solving the damage evolution, the displacement increment is expressed in terms of the increment of the phase variable. The relation between these two increments enables a prediction of the displacement and the active energy based on the increment of the phase variable. Hence, the maximum number of staggered iterations is reduced, and the computational efficiency is improved. We present three fast staggered schemes by fixing the first invariant, second invariant, or both invariants of the stress, denoted by S1, S2, and S3 schemes. The performance of the schemes is then verified by comparing with the standard staggered scheme through three benchmark examples, i.e., tensile, shear, and L-shape panel tests. The results exhibit that the force–displacement relations and the crack patterns computed using the fast schemes are consistent with the ones based on the standard staggered scheme. Moreover, the proposed S1 and S3 schemes can largely reduce the maximum number of staggered iterations and total cpu time in all benchmark tests. The S2 scheme performs comparably except in the L-shape panel test, where the underlying assumption is violated in the region close to the crack.

Combined diffused material interface and hybrid phase-field model for brittle fracture in heterogeneous composites

In this article, we propose a novel approach for modeling brittle fracture in heterogeneous composites using a combined diffused material interface method and a hybrid phase-field model (PFM). We have used the diffused material interface method to derive analytical expressions for the regularized material properties across the material interfaces that are treated as an input to the PFM for the simulation of brittle fractures in composites. The use of diffused material interface method regularizes the material properties in such a way that one can avoid the discontinuity in stress across the material interface and thus bypass the requirement of explicit tracking of material interfaces for finite element implementation of the hybrid PFM. We have implemented the proposed combined model using a recently developed open-source finite element toolbox, Gridap in Julia. We have considered tests on a single circular fiber inclusion under tension, a bi-material rectangular plate under tension, single square inclusion under tensile loading, a notched composite beam with a central notch under symmetric three-point bending, and a rectangular plate with multiple fibers under tensile loading to show how the proposed combined model can be used to simulate brittle fracture in composites.

Nonlinear field-split preconditioners for solving monolithic phase-field models of brittle fracture

One of the state-of-the-art strategies for predicting crack propagation, nucleation, and interaction is the phase-field approach. Despite its reliability and robustness, the phase-field approach suffers from burdensome computational cost, caused by the non-convexity of the underlying energy functional and a large number of unknowns required to resolve the damage gradients. In this work, we propose to solve such nonlinear systems in a monolithic manner using the Schwarz preconditioned inexact Newton (SPIN) method. The proposed SPIN method leverages the field split approach and minimizes the energy functional separately with respect to displacement and the phase-field, in an additive and multiplicative manner. In contrast to the standard alternate minimization, the result of this decoupled minimization process is used to construct a preconditioner for a coupled linear system, arising at each Newton’s iteration. The overall performance and the convergence properties of the proposed additive and multiplicative SPIN methods are investigated by means of several numerical examples. A comparison with widely-used alternate minimization is also performed showing a significant reduction in terms of execution time. Moreover, we also demonstrate that this reduction grows even further with increasing problem size.

A mode-adjustable phase-field model for brittle fracture by regulating distortional crack driving energy

A mode-adjustable phase-field model is proposed by introducing a weight parameter to regulate the relative contribution of the volumetric and distortional crack driving energy. Altering the weight parameter results in the change of cracking mode. The weight parameter can be considered as a material parameter and determined through the comparison of experimental and simulated results. The proposed phase-field model is employed to simulate the fracture experiments for polymethylmethacrylate and Al 7075-T651 aluminum. Applying the experimentally determined parameter, the mode-adjustable phase-field model provides better simulations than the classical phase-field model for both materials. Afterwards, a shear test of a single edge notched plate is simulated and the results show that for the volumetric-deviatoric split method, as the weight parameter increases, the failure pattern during crack propagation exhibits a transition from the mode-I dominant failure to the mode-II dominant failure gradually. On the contrary, a mode-I dominant cracking always occurs with the initial crack deflection angle of about 70.5°, regardless of the value of weight parameter for the spectral decomposition split method. To summarize, the proposed mode-adjustable phase-field model can reflect the difference of crack driving mechanism, which is helpful for capturing and understanding the fracture characteristics of diverse materials.

Phase-field modeling of brittle fracture in heterogeneous bars

We investigate phase-field modeling of brittle fracture in a one-dimensional bar featuring a continuous variation of elastic and/or fracture properties along its axis. Our main goal is to quantitatively assess how the heterogeneity in elastic and fracture material properties influences the observed behavior of the bar, as obtained from the phase-field modeling approach. The results clarify how the elastic limit stress, the peak stress and the fracture toughness of the heterogeneous bar relate to those of the reference homogeneous bar, and what are the parameters affecting these relationships. Overall, the effect of heterogeneity is shown to be strictly tied to the non-local nature of the phase-field regularization. Finally, we show that this non-locality may amend the ill-posedness of the sharp-crack problem in heterogeneous bars where multiple points compete as fracture locations.

Three-dimensional phase-field modeling of brittle fracture using an adaptive octree-based scaled boundary finite element approach

In this paper, an adaptive phase-field approach is proposed for three-dimensional fracture modeling in brittle materials. The scaled boundary finite element method (SBFEM) is used to solve the phase-field and elasticity equations in a staggered scheme. This is motivated by the ability of the SBFEM to handle polyhedral elements with hanging nodes straightforwardly. Such elements occur in octree meshes, which facilitate rapid transitions in element size and lend themselves to adaptive refinement. The SBFEM also provides an energy-based error indicator, which follows directly from the semi-analytical solution approach and does not require stress recovery. The error indicator is used to ensure the solution accuracy for both fields and combined with a criterion based on phase-field evolution. The computational cost associated with 3D fracture modeling is further reduced by exploiting the geometric similarity of cells occurring in balanced octree meshes. To validate the proposed method, several classical benchmark problems are solved, resulting in efficient and robust solutions compared to the literature.