Brittle Fracture Model Research Articles

A meshfree peridynamic model for brittle fracture in randomly heterogeneous materials

In this work we aim to develop a unified mathematical framework and a reliable computational approach to model the brittle fracture in heterogeneous materials with variability in material microstructures, and to provide statistical metrics for quantities of interest, such as the fracture toughness. To describe the material responses such as the nucleation and growth of fractures in a uniform setting, we consider the peridynamics model. In particular, a stochastic state-based peridynamic model is developed, where the micromechanical parameters are modeled by a finite-dimensional random field, e.g., a combination of truncated random variables determined by the Karhunen–Loève decomposition or the principal component analysis (PCA). To solve this stochastic peridynamic problem, probabilistic collocation method (PCM) is employed to sample the random field that represents the micromechanical parameters. For each sample, the corresponding peridynamic problem is solved by an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and verify the convergence rate with a number of benchmark problems. The proposed scheme not only possesses the asymptotic compatibility spatially but also achieves an algebraic or sub-exponential convergence rate in the parametric random space when the number of collocation points grows. Finally, to validate the applicability of this approach on real-world fracture problems, we consider the problem of crystallization toughening in glass-ceramic materials, in which the material at the microstructural scale contains both amorphous glass and crystalline phases. The proposed stochastic peridynamic solver is employed to capture the crack initiation and its growth of the glass-ceramics with different crystal volume fractions, and the averaged fracture toughness are also calculated accordingly. The numerical estimates of fracture toughness show good consistency with data from experimental measurements.

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.

Sea ice floe size: its impact on pan-Arctic and local ice mass and required model complexity

Abstract. Sea ice is composed of discrete units called floes. Observations show that these floes can adopt a range of sizes spanning orders of magnitude, from metres to tens of kilometres. Floe size impacts the nature and magnitude of interactions between the sea ice, ocean, and atmosphere including lateral melt rate and momentum and heat exchange. However, large-scale geophysical sea ice models employ a continuum approach and traditionally either assume floes adopt a constant size or do not include an explicit treatment of floe size. In this study we apply novel observations to analyse two alternative approaches to modelling a floe size distribution (FSD) within the state-of-the-art CICE sea ice model. The first model considered is a prognostic floe size–thickness distribution where the shape of the distribution is an emergent feature of the model and is not assumed a priori. The second model considered, the WIPoFSD (Waves-in-Ice module and Power law Floe Size Distribution) model, assumes floe size follows a power law with a constant exponent. We introduce a parameterisation motivated by idealised models of in-plane brittle fracture to the prognostic model and demonstrate that the inclusion of this scheme enables the prognostic model to achieve a reasonable match against the novel observations for mid-sized floes (100 m–2 km). While neither FSD model results in a significant improvement in the ability of CICE to simulate pan-Arctic metrics in a stand-alone sea ice configuration, larger impacts can be seen over regional scales in sea ice concentration and thickness. We find that the prognostic model particularly enhances sea ice melt in the early melt season, whereas for the WIPoFSD model this melt increase occurs primarily during the late melt season. We then show that these differences between the two FSD models can be explained by considering the effective floe size, a metric used to characterise a given FSD. Finally, we discuss the advantages and disadvantages to these different approaches to modelling the FSD. We note that although the WIPoFSD model is unable to represent potentially important features of annual FSD evolution seen with the prognostic model, it is less computationally expensive and produces a better fit to novel FSD observations derived from 2 m resolution MEDEA imagery, possibly making this a stronger candidate for inclusion in climate models.

A smoothed particle hydrodynamics approach for phase field modeling of brittle fracture

Fracture is a very challenging and complicated problem with various applications in engineering and physics. Although it has been extensively studied within the context of mesh-based numerical techniques, such as the finite element method (FEM), the research activity within the Smoothed Particle Hydrodynamics (SPH) community remains scarce. SPH is a particle-based numerical method used to discretize equations of continuum media. Its meshfree nature makes it ideal to simulate fracture scenarios that involve extreme deformations. However, to model fracture, SPH researchers have mostly relied on ad-hoc empirical local damage models, cohesive zone approaches, or pseudo-spring models, which come with a set of drawbacks and limitations. On the other hand, phase field models of brittle fracture have recently gained popularity in academic circles and provide significant improvements compared to previous approaches. These improvements include the derivation from fundamental fracture theories, the introduction of non-locality, and the ability to model multiple crack initiation, propagation, branching, and coalescence, in situations where no prior knowledge of the crack paths is available. Nevertheless, phase field for fracture has not been studied within SPH. In this proof-of-concept paper we develop and implement a phase field model of brittle fracture within the context of SPH. Comprehensive mathematical and implementation details are provided, and several challenging numerical examples are computed and illustrate the proposed method’s ability to accurately and efficiently simulate complex fracture scenarios.

Phase‐field modeling of brittle fracture along the thickness direction of plates and shells

AbstractThe prediction of fracture in thin‐walled structures is decisive for a wide range of applications. Modeling methods such as the phase‐field method usually consider cracks to be constant over the thickness which, especially in load cases involving bending, is an imperfect approximation. In this contribution, fracture phenomena along the thickness direction of structural elements (plates or shells) are addressed with a phase‐field modeling approach. For this purpose, a new, so called “mixed‐dimensional” model is introduced, which combines structural elements representing the displacement field in the two‐dimensional shell midsurface with continuum elements describing a crack phase‐field in the three‐dimensional solid space. The proposed model uses two separate finite element discretizations, where the transfer of variables between the coupled two‐ and three‐dimensional fields is performed at the integration points which in turn need to have corresponding geometric locations. The governing equations of the proposed mixed‐dimensional model are deduced in a consistent manner from a total energy functional with them also being compared to existing standard models. The resulting model has the advantage of a reduced computational effort due to the structural elements while still being able to accurately model arbitrary through‐thickness crack evolutions as well as partly along the thickness broken shells due to the continuum elements. Amongst others, the higher accuracy as well as the numerical efficiency of the proposed model are tested and validated by comparing simulation results of the new model to those obtained by standard models using numerous representative examples.

Adaptive modelling of dynamic brittle fracture - a combined phase field regularized cohesive zone model and scaled boundary finite element approach

Adaptive modelling of dynamic brittle fracture - a combined phase field regularized cohesive zone model and scaled boundary finite element approach

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.

Correction to: A directional-dependent localized mass-field damage model for anisotropic brittle fracture

Correction to: A directional-dependent localized mass-field damage model for anisotropic brittle fracture

U-notch fracture in additively manufactured ABS specimens under symmetric three-point bending

This paper aims to investigate the fracture behavior of U-notched semi-circular bend (USCB) specimens fabricated from the Acrylonitrile Butadiene Styrene (ABS) using the fused deposition modelling (FDM) method under opening-mode loading. The notched specimens are manufactured with different notch tip radii and various unidirectional layup configurations for which the fracture loads are experimentally acquired by conducting fracture tests under symmetric three-point bending. Some additional tests are also performed on cracked and plain tensile specimens to obtain the fracture toughness and tensile properties of the ABS material. With the goal of estimating the fracture loads theoretically, the recently proposed virtual isotropic material concept (VIMC) is linked to three different brittle fracture models, namely the point-stress (PS), mean-stress (MS), and averaged strain energy density (ASED) criteria. Comparing with the experimental results, it is shown that the VIMC-MS criterion is generally the best model among the three ones, while VIMC-PS criterion gives fair results and VIMC-ASED criterion is excellent only for larger notch tip radii. The crack growth path of the mentioned samples as well as their type of failure is also evaluated. It is revealed that the unidirectional USCB samples with 0, 30, and 45 degrees layup configurations experience inter/cross-laminar fracture, while those with 60 and 90 degrees layups experience inter-laminar fracture.

Phase-field modelling of brittle fracture with Smoothed Radial Point Interpolation Methods

The phase-field modelling of brittle fracture is an efficient tool for the modelling of complex cracks, since it is capable to predict crack initiation and branching, as well as the merging of cracks. While there are a number of papers in the literature devoted to the application of different meshfree methods to the phase-field modelling of fracture, this paper is, to the authors knowledge, the first application of the methods of the Smoothed Point Interpolation Methods (SPIM) family. The smoothing operation, that is used to approximate the displacements gradient in standard SPIM strategies, is here extended to the additional variable of a phase-field model. A number of numerical simulations, performed considering different kind of smoothing domains and support nodes selection strategies, illustrates the characteristics of the proposed approach.

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.

A general and fast convolution-based method for peridynamics: Applications to elasticity and brittle fracture

A general and fast convolution-based method (FCBM) for peridynamics (PD) is introduced. Expressing the PD integrals in terms of convolutions and computing them by Fast Fourier Transform (FFT), the computational complexity of PD models drops from O(N2) to O(Nlog2N), with N being the number of discretization nodes. Initial neighbor identification and storing neighbor information is not required, and this means memory allocation scales with O(N) instead of O(N2), common for existing methods. FCBM is applicable to bounded domains with arbitrary shapes and boundary conditions via an “embedded constraint” (EC) approach. The formulation is shown for certain bond-based and state-based, linear and nonlinear, PD models of elasticity and dynamic brittle fracture, as applications. The method is verified on a 3D elastostatic problem and it is shown that the FCBM-PD reduces the computational time from days to hours and from years to days, compared with the original meshfree discretization for PD models. Large-scale computations of PD models are feasible with the new method, and its versatility is demonstrated by simulating, with ease, the difficult problem of cascading crack branching in a brittle plate.

A directional-dependent localized mass-field damage model for anisotropic brittle fracture

A directional-dependent localized mass-field damage model for anisotropic brittle fracture

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.

Employing phase‐field descriptions of cohesive zone placements in cohesive fracture simulations

AbstractCohesive zone models suffer from the fundamental problem that potential crack paths must be determined beforehand. Phase‐field models for (brittle) fracture, that have recently become immensely popular, are able to naturally handle essentially arbitrary crack patterns. However, due to their energetic nature, phase‐field approaches have difficulties to predict, for example, crack nucleation in consequence of their lack of strength criteria which are naturally incorporated in the cohesive zone model. In this contribution, an alternative approach is presented that allows the embedding of established (cyclic) cohesive laws into phase‐field modeling. Our work is conceptually in line with the ideas proposed by Verhoosel and de Borst in 2013. Since this novel approach to cohesive fracture is still in its infancy, many open challenges remain, of which the following are addressed in this article: (i) A strict separation between the description of the cohesive zone and the cohesive law itself is realized, (ii) the interplay between the different physical length‐scales inherent to the formulation is carefully investigated in order to quantify the parameters introduced in the phase‐field description, and (iii) an alternative finite element treatment is proposed, implemented, and tested that avoids spurious solutions for unstructured meshes. In this context, fatigue crack growth is simulated as a quantitative benchmark problem relevant to engineering applications.

Investigation of notch effects on load-bearing capacity of AA7075-AA7075 friction-stir welded joints under mixed mode I/II loading

In this research, mixed mode I/II fracture of AA7075-AA7075 friction-stir welded (FSWed) joints weakened by V-notches is studied. This study consists of two parts: experimental and theoretical. In the experimental part, the load-bearing capacity (LBC) of rectangular specimens made of welded AA7075-T6 aluminum alloy containing a central V-shaped notch under mixed mode I/II loading is measured. Due to the elastic–plastic behavior of the specimens and noticeable deformations at the notch neighborhood, prediction of fracture of the specimens under test using nonlinear fracture mechanics criteria is complex and time consuming. As a result, in the theoretical part, using the Equivalent Material Concept (EMC) in combination with two brittle fracture models (mean stress, MS, criterion and maximum tangential stress, MTS, criterion), the LBC of the samples tested is estimated. By comparing the experimental results with the theoretical ones, it is clear that both theoretical criteria used in combination with EMC are well able to predict the LBC of the notched specimens. Also, based on the fracture tests performed and the analysis of the samples in a commercial finite element software, it is observed that the size of the plastic zone in the vicinity of notch increases with increasing the share of mode II, i.e. from pure mode I to pure mode II. Observed is also that the enlargement of the plastic zone increases the LBC of the samples.

A strain gradient brittle fracture model based on two-scale asymptotic analysis

Based on a two-scale asymptotic analysis procedure, we rigorously derive a new macroscopic brittle fracture model for materials with micro-cracks inside. Strain gradient effects are naturally involved in the model in the absence of any phenomenological assumptions. Coefficients of the fracture criterion are expressed in form of derivatives of the fourth- and sixth-order effective elastic tensors that are explicitly related to solutions of the first order unit cell problems. The model is numerically implemented into a finite element code. Abilities of the model to predict microstructure size effects and strain gradient effects on macro fracture behaviors are confirmed by numerical simulations. The model is also validated against experimental measurements.

Mathematical Model of Brittle Fracture of a Cracked Part

Mathematical Model of Brittle Fracture of a Cracked Part

On a new high order phase field model for brittle and cohesive fracture: numerical efficiency, length scale convergence and crack kinking

Being able to seamlessly deal with complex three dimensional crack patterns like branching and merging, phase field models (PFMs) are promising in the computational modelling of fracture of solids. Regarding the damage field, if only its second order derivative is present in the governing equation we have a second order PFM and if its fourth order derivative is also present we have a fourth order PFM. Previous studies have demonstrated that fourth order PFMs, with its smoother damage profile, are better in convergence and able to model strong anisotropic fracture energies. However, previous fourth order PFMs were developed only for brittle fracture and do not embed the material strength directly in the formulation. This paper presents a fourth order phase field regularised cohesive zone model (fourth order PF-CZM) for both brittle fracture and quasi-brittle fracture. We present a semi-analytical approach to compute the coefficients of the rational degradation function used in the fourth order PF-CZM. The proposed model is tested with multiple benchmark problems for mode I and mixed-mode fracture, and the results demonstrate that the fourth order PF-CZM: (1) provides results independent of the length scale and (2) is more efficient than its second order counterpart. The proposed model was then applied to modelling strong anisotropic fracture energies. A detailed study on sawtooth like crack patterns, which is a signature of strongly anisotropic fracture, is carried out with focus on periodicity and length scale sensitivity.

Phase‐field fracture simulations of a four‐point bending test

AbstractPhase‐field fracture simulations have been established to simulate crack propagation in solid structures. The method employs a variational framework which has been proven to converge to Griffith' classical model for brittle fracture. Here we investigate the predictiveness of phase‐field fracture through a four‐point bending test in a mixed mode condition, and we discuss the possibilities to perform such simulation with the finite element software ABAQUS. The comparison with the experiments allows estimating the quality of the method with respect to failure loads, crack initiation angles, propagation path, and fracture surface.