An adaptive singular finite element in nonlinear fracture mechanics

In the case of nonlinear fracture mechanics the type of singularity induced by the crack tip is commonly not known. This results in a poor approximation of the near crack tip fields in a finite element setting and induces so called spurious—or residual—discrete material forces in the vicinity of the crack tip. Thus the numerical calculation of the crack driving material force in nonlinear fracture is often not that precise as in linear elasticity where we can use special crack tip elements and/or path independency. To overcome this problem we propose an adaptive singular element, which adapts automatically to the type of singularity. The adaption is based on an optimisation procedure using a variational principle.     

A generalized phase field multiscale finite element method for brittle fracture

A generalized multiscale finite element method is introduced to address the computationally taxing problem of elastic fracture across scales. Crack propagation is accounted for at the microscale utilizing phase field theory. Both the displacement-based equilibrium equations and phase field state equations at the microscale are mapped on a coarser scale. The latter is defined by a set of multinode coarse elements, where solution of the governing equations is performed. Mapping is achieved by employing a set of numerically derived multiscale shape functions. A set of representative benchmark tests is used to verify the proposed procedure and assess its performance in terms of accuracy and efficiency compared with the standard phase field finite element implementation.     

A finite element analysis of fracture initiation in ductile/brittle periodically layered composites

A near-tip plane strain finite element analysis of a crack terminating at and normal to the interface in a laminate consisting of alternate brittle and ductile layers is conducted under mode-I loading. The studies are carried out for a system representing steel/alumina composite laminate. The Gurson constitutive model, which accounts for the ductile failure mechanisms of microvoid nucleation, growth and coalescence, is employed within the framework of small deformation plasticity theory. Evolution of plastic zone and damage in the ductile layer is monitored with increasing load. High plastic strain localization and microvoid damage accumulation are found to occur along the brittle/ductile interface at the crack-tip. Fracture initiation in the ductile phase is predicted and the conditions for crack renucleation in the brittle layer ahead of the crack are established for the system under consideration. Ductile fracture initiation has been found to occur before plasticity spreads in multiple ductile layers. Effects of material mismatch and yield strength on the plastic zone evolution are briefly discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

A hybrid adaptive finite element phase-field method for quasi-static and dynamic brittle fracture

The phase-field approach has unique advantages in describing fracture phenomena, which has received extensive attention in the past decade. Nevertheless, the phase-field modeling of fracture is computationally demanding, due to the high temporal-spatial resolution required for crack tracking. In this contribution, a novel hybrid adaptive finite element phase-field method (ha-PFM) is developed to solve brittle fracture problems under quasi-static and dynamic loading. ha-PFM can dynamically track the propagation of the cracks and adaptively refine the meshes based on a novel crack tip identification strategy. Afterward, the refined meshes in the noncrack progression region are reconverted into coarse meshes. This scheme prominently reduces the computational cost, eg, CPU time and memory usage. Unlike the previous adaptive phase-field method, multilevel hybrid triangular and quadrilateral elements were developed to discretize the computational domain, which eliminates hanging nodes and ensures that the meshes in the vicinity of the crack tip are highly isotropic. Several representative benchmarks containing quasi-static and dynamic fracture were reinvestigated with ha-PFM, and its excellent performance is substantiated by comparison with the standard phase-field method and literature results.     

A finite element ductile failure simulation method using stress-modified fracture strain model

This paper proposes a new method to simulate ductile failure using finite element analysis based on the stress-modified fracture strain model. A procedure is given to determine the stress-modified fracture strain as a function of the stress triaxiality from smooth and notched bar tensile tests with FE analyses. For validation, simulated results using the proposed method are compared with experimental data for cracked bar (tensile and bend) tests, extracted from API X65 pipes, and for full-scale burst test of gouged pipes, showing overall good agreements. Advantages in the use of the proposed method for practical structural integrity assessment are discussed.     

A conic-section simulation analysis of two-dimensional fracture problems using the finite element method

A conic-section simulation analysis to determine the stress intensity factors for fracture mechanics problems of practical interest using the finite element method is presented. The method makes use of elliptic displacement functions which are satisfied by the introduction of an equivalent ellipse obtained through first simulating the actual crack surface displacements as a part of a parabola or a hyperbola. Unlike other finite element approaches that incorporate no special crack-tip treatment, the present approach requires neither extremely small finite elements in the vicinity of the crack tip nor the computation of several strain energies. The cases examined include not only problems of the opening mode (I) or the edge-sliding mode (II), but also the combined modes of crack deformation.

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Résumé On présente une analyse par simulation permettant de déterminer les facteurs d'intensité de contraintes pour des problèmes de mécanique de rupture d'intérêt pratique, en utilisant la méthode des éléments finis. On recourt à des fonctions de déplacement elliptique, qui sont satisfaites par l'introduction d'une ellipse équivalente, obtenue en assimilant les déplacements réels en surface de la rupture à une portion de parabole ou d'hyperbole. Au contraire des autres approches par éléments finis qui ne prévoient pas un traitement particulier de l'extrémité de la fissure, la méthode proposée ne requiert ni de mailles extrêmement fines au voisinage de cette extrémité, ni de nombreux calculs d'énergies de déformation. Les cas examinés ne se limitent pas aux problèmes d'ouverture de mode I ou de mode II, mais couvrent aussi des modes combinés déformation de la fissure.

     

A criterion study for non-singular stress concentrations in brittle or quasi-brittle materials

In engineering applications, it is difficult to avoid the non-singular stress concentrations that often play an important role in structure designs. The simplest engineering strength criteria are in general not appropriate due to their incapacity in dealing with important size effect induced by stress gradients. In this paper, we present first a simple experimentation, which consists of plates with a central hole under uniaxial tensile loading, showing important size effect. Second, numerous criteria, including commonly used engineering criteria, crack initiation criteria based on the finite fracture mechanics, or cohesive criteria, were adapted to fit the experimental results. We found that most of these criteria, including criteria with a single material parameter and those with two material parameters, are not suitable for fracture prediction of materials under non-singular stress concentrations. It seems that three material parameters would be the minimum to establish an adequate fracture criterion for arbitrary stress concentrations. By analysing the energy dissipation of micro-crack bands under different stress concentrations, we proposed a new fracture criterion with three material parameters based on the finite fracture mechanics. It is shown that this criterion can provide accurate critical remote loads comparing with experimental data. We believe that the three parameter concept is physically reasonable and can be used in establishing fracture criteria in both the cases of singular and non-singular stress concentrations.     

A failure criterion for brittle and quasi-brittle materials under any level of stress concentration

In this paper, we present a new criterion to predict the crack initiation under quasi-static loads from a geometrical weakness presenting an arbitrary stress concentration in brittle or quasi-brittle materials. Three material parameters were used in the establishment of the criterion, namely the ultimate stress σ_c, the critical energy release rate for crack growth G_c and the critical energy release rate for fracture under uniform uniaxial tension G_u. The use of these two critical energy release rates was justified by the observation of the fracture surfaces under different stress concentrations. The proposed three parameters’ concept enables to take the different stress concentration levels into account, thus provides a unified criterion to predict crack initiation for any stress concentration, whatever it is singular or regular. Numerous experimental studies were selected to verify the accuracy and efficiency of the criterion. It was shown that the proposed criterion is physically reasonable, highly accurate and easy to apply. It can be used in crack initiation prediction of engineering structures made of brittle or quasi-brittle materials.     

A singular edge-based smoothed finite element method (ES-FEM) for crack analyses in anisotropic media

The recently developed edge-based smoothed finite element method (ES-FEM) is extended to fracture problems in anisotropic media using a specially designed five-node singular crack-tip (T5) element. In the formulation of singular ES-FEM, only the assumed displacement values (not the derivatives) on the boundaries of the smoothing domains are needed. Thus, a layer of T5 crack-tip element is devised to construct “singular” shape functions via a simple point interpolation with a fractional order basis, without mapping procedure. The effectiveness of the present singular ES-FEM is demonstrated by intensive examples for a wide range of degrees of anisotropy.     

Mesh design in finite element analysis of post-buckled delamination in composite laminates

The role of mesh design in the post-buckling analysis of delamination in composite laminates is addressed in this paper. The determination of the strain energy release rate (SERR) along the crack front is central to the analysis. Frequently, theoretical analysis is limited to treatment of the problem in two dimensions, since considerable complexity is encountered in extending the analysis to three dimensions. However, many practical problems of embedded delamination in composite laminates are inherently three-dimensional in nature. Although in such cases, the finite element (FE) method can be employed, there are some issues that must be examined more closely to ensure physically realistic models. One of these issues is the effect of mesh design on the determination of the local SERR along the delamination front. There are few studies that deal with this aspect systematically. In this paper, the effect of mesh design in the calculation of SERR in two-dimensional (2D) and three-dimensional (3D) FE analyses of the post-buckling behavior of embedded delaminations is studied and some guidelines on mesh design are suggested. Two methods of calculation of the SERR are considered: the virtual crack closure technique (VCCT) and crack closure technique (CCT). The 2D analyses confirm that if the near-tip mesh is symmetric and consists of square elements, then the evaluation of the SERR is not sensitive to mesh refinement, and a reasonably coarse mesh is adequate. Despite agreement in the global post-buckling response of the delaminated part, the SERR calculated using different unsymmetrical near-tip meshes could be different. Therefore, unsymmetrical near-tip meshes should be avoided, as convergence of the SERR with mesh refinement could not be assured. While the results using VCCT and CCT for 2D analyses agree well with each other, these techniques yield different quantitative results when applied to 3D analyses. The reason may be due to the way in which the delamination growth is modeled. The CCT allows simultaneous delamination advance over finite circumferential lengths, but it is very difficult to implement and the results exhibit mesh dependency. Qualitatively, however, the two sets of results show similar distributions of Mode I and Mode II components of the SERR. This is fortunate, since the VCCT is relatively easy to implement.     

Identification of CTOA and fracture process parameters by drop weight test and finite element simulation

This paper presents a new technique that is able to predict ductile fracture propagation occurrences in large metallic structures, by means of an appropriate application of the finite element modelling. This technique takes account of a cohesive zone in the vicinity of the crack tip, where a nodal release technique is implemented. Two parameters, governing the process zone of the material under investigation, have to be determined: the process zone dimension (named “Δ distance”) and the critical value of crack tip opening angle (CTOA). CTOA_C can be determined through an experimental laboratory procedure two specimen CTOA test (TSCT) that is already known and used by researchers who study fracture propagation on pipelines [Demofonti G, et al. Step-by-step procedure for the two specimen CTOA test. In: Proceedings of the Second International Conference on Pipeline Technology, Ostend, vol. II. 1995]. The second parameter required, Δ distance, is determined minimizing the differences of Finite Element results towards experimental data of an instrumented impact test (drop weight tear test). Some interesting improvements, concerning distinction between the initiation energy and the propagation energy accounted in TSCT procedure, are also discussed, in order to successfully extend its use to both high strength and high toughness steels.     

Use of finite element computer programs in fracture mechanics

Since the theoretical stresses and strains at the tip of a V-notched crack in an elastic continuum are infinite, the question arises as to the accuracy of strain energy as calculated from finite element computer programs for systems containing such a crack. Two geometries for which analytical solutions are available were analyzed using a plane stress finite element computer program. Results show that accuracy in both cases depended upon proper selection of a grid network. Several methods of calculating stress intensity factors are discussed. Application of the finite element computer program in the analysis of fracture in solid propellant rocket motor cartridge or grain is included.

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Zusammenfassung Die theoretischen Spannungen und Dehnungen an der Spitze eines V-förmigen Risses in einem elastischen Kontinuum sind unendlich groß. Es stellt sich die Frage der Genauigkeit der, für Systeme mit solchen Rissen, nach dem Verfahren der endlichen Elementen errechneten Dehnungsenergie.Zwei geometrische Formen, für welche analytische Ergebnisse vorlagen, wurden an Hand eines Rechenprogramms für die Ermittlung des planen Spannungszustandes durch endliche Elemente untersucht.Die Ergebnisse zeigen, daß in beiden Fällen die Genauigkeit von der Wahl eines passenden Netzes abhängt.Es werden verschiedene Verfahren zur Bestimmung der Spannungsintensitätsfaktoren besprochen; die Anwendung des Rechenprogramms zur Ermittlung der Bruchbedingungen in Festbrennstoffelementen für Raketen wird behandelt.

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Résumé Dans un continuum élastique, les contraintes et les déformations à l'extrémité d'une fissure en Vé sont théoriquement infinies. La question se pose donc de savoir quel est le degré de précision que l'on peut attendre du calcul à l'ordinateur par éléments finis de l'énergie de déformation correspondant à des systèmes où se rencontrent de telles fissures.On a analysé deux types de géométries différentes, pour lesquelles existent des solutions analytiques, en utilisant un programme d'ordinateur pour l'étude par éléments finis de l'état plan de tension.Les résultats montrent que, daps les deux cas, la précision dépend du choix du réseau le plus adéquat.Diverses méthodes de calcul des facteurs d'intensité des contraintes sont discutées. On traite de l'application des programmes de calcul par ordinateur des élements finis à l'analyse des conditions de rupture dans les cartouches ou éléments de combustible solide pour fusées.

     

On the role of stress fluctuations in brittle fracture

Cracks in random stress fields are assumed to be originated in regions with high local tension. As a legacy of this special location, additional local tractions opening the crack in its centre are developed even in self-equilibrating stress fields. As the crack becomes a mesocrack it will deviate its path to meet the regions with higher possible local tension. The necessary statistical properties of the microcrack-generated random stress field can be calculated using the dipole asymptotics to approximate the stresses generated by each microcrack. The microcracks are assumed to be noninteracting and surrounded by nonintersecting excluded volumes. For the case of spherical excluded volumes the correlation radius is found to be less than the microcrack radius, which suggests that the stresses acting on each microcrack can be assumed to be statistically independent. In brittle fracture under uniaxial tension the effect of the stress fluctuations is shown to be able to significantly reduce the macroscopic strength. In fracture of brittle materials under uniaxial compression wing cracks are developed which, in real 3-D situations, cannot grow extensively and therefore cannot themselves cause failure. Instead, they induce stress fluctuations which generate mesocracks growing towards compression in such a way as to avoid the wing cracks. Hence, only stresses outside excluded volumes around the wing cracks will affect the mesocrack growth. These stresses have positive mean even if the full stress field is self-equilibrating. This results in a background tension acting perpendicular to the compression axis, amplifying the mesocrack growth and eventually causing failure. The growth and opening of mesocracks results in a specific dependence between dilatancy, i.e. inelastic increase of the sample volume, and the applied compressive stress. This dependence has a universal nature independent of the particular model of wing cracks. It corresponds well to the data of uniaxial compressive tests on 4 samples of Oshima granite (Sano et al. 1981) despite markedly different loading rates and resulted strengths. This revised version was published online in July 2006 with corrections to the Cover Date.     

Probabilistic fracture mechanics by using Monte Carlo simulation and the scaled boundary finite element method

A numerical technique to model the effect of uncertainties in the crack geometry on the reliability of cracked structures is presented. The shape sensitivity analysis of stress intensity factors to the crack size and orientation is performed by using the scaled boundary finite element method (SBFEM). Only a single boundary mesh is required. The varying crack size and orientation are represented by simply moving the scaling center and without the need for remeshing. The reliability assessment is performed by Monte Carlo simulations. Numerical examples are analyzed to verify the accuracy and demonstrate the efficiency and simplicity of the proposed technique.     

The extended finite element method in thermoelastic fracture mechanics

The extended finite element method (XFEM) is applied to the simulation of thermally stressed, cracked solids. Both thermal and mechanical fields are enriched in the XFEM way in order to represent discontinuous temperature, heat flux, displacement, and traction across the crack surface, as well as singular heat flux and stress at the crack front. Consequently, the cracked thermomechanical problem may be solved on a mesh that is independent of the crack. Either adiabatic or isothermal condition is considered on the crack surface. In the second case, the temperature field is enriched such that it is continuous across the crack but with a discontinuous derivative and the temperature is enforced to the prescribed value by a penalty method. The stress intensity factors are extracted from the XFEM solution by an interaction integral in domain form with no crack face integration. The method is illustrated on several numerical examples (including a curvilinear crack, a propagating crack, and a three‐dimensional crack) and is compared with existing solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

Estimations of creep fracture mechanics parameters for through-thickness cracked cylinders and finite element validation

This paper compares engineering estimation schemes of C* and creep crack opening displacement (COD) for cylinders with circumferential and axial through‐thickness cracks at elevated temperatures with detailed 3D elastic‐creep finite element results. Engineering estimation schemes include the GE/EPRI method; the reference stress (RS) method where the reference stress is defined based on the plastic limit load; and the enhanced reference stress (ERS) method where the reference stress is defined based on the optimised reference load, recently proposed by the authors. Systematic investigations are made not only on the effect of creep‐deformation behaviour on C* and creep COD, but also on effects of the crack location, the cylinder geometry, the crack length and the loading mode. Comparison of the finite element (FE) results with engineering estimations provides that for idealised power law creep, estimated C* and COD rate results from the GE/EPRI method agree best with FE results, suggesting that published plastic influence functions for plastic J and COD for through‐thickness cracked cylinders are reliable. For general creep‐deformation laws where either primary or tertiary creep is important and thus the GE/EPRI method is hard to apply, on the other hand, the ERS method provides more accurate and robust estimations for C* and COD rate than the reference stress method. As these two methods differ only in the definition of the reference stress, the ERS method maintains benefits of the reference stress method in terms of simplicity, but improves accuracy of the estimated J, C* and COD results.     

Failure analysis and finite element simulation of deformation and fracture of an exploded CNG fuel tank

This paper reports the analysis and simulation of the catastrophic failure of a compressed natural gas (CNG) fuel tank. The initial analyses of the deformation and cracking patterns, along with the observed fractographic features, were indicative of an internal gaseous combustion. Accordingly, a set of transient-dynamic elasto-plastic finite element (FE) analyses was carried out to simulate the structural response of the tank to a special type of combustion-induced dynamic pressure. The FE model was composed of 3D brick elements equipped with interface cohesive elements for crack growth analysis. Excellent agreements were found between the final simulation results and the observed deformation and fracture patterns. The simulation results clearly revealed that the observed failure characteristics, like the overall asymmetric deformation and fracture patterns, initiation and partial growth of parallel cracks at the same section, multiple cracking at the neck, and the self-similar growth of the main axial crack were all caused by traveling of a deflagration-induced sonic pressure wave from the neck towards the bottom of the tank. Finally, a comparison was made between the characteristics of deflagration-induced and detonation-induced deformation and fracture behaviors of closed-end cylinders.     

A geometric nonlinear triangular finite element with an embedded discontinuity for the simulation of quasi-brittle fracture

This paper is aimed to model the appearance and evolution of discrete cracks in quasi-brittle materials using triangular finite elements with an embedded interface in a geometric nonlinear setting. The kinematics for the discontinuous displacement field is presented and the standard variational formulation with respect to the reference configuration is extended to a body with an internal discontinuity. Special attention is paid to the algorithmic treatment. The discontinuity is modeled by additional global degrees of freedom and the continuity of the displacements across the element boundaries is enforced. Finally, representative numerical examples for mode-I and mixed-mode fracture, namely a tension test, different three-point bending tests and a single edge notched beam with structured and unstructured finite element meshes are discussed to study the evolving crack pattern and to show the ability of the model.     

Extended finite element fracture analysis of a cracked isotropic shell repaired by composite patch

An extended finite element method (XFEM) is developed to study fracture parameters of cracked metal plates and tubes that are repaired on top of the crack with a composite patch. A MATLAB® stand‐alone code is prepared to model such structures with eight‐noded doubly curved shell elements in the XFEM framework. Crack trajectory studies are performed for a diagonally cracked panel under fatigue loading. Verification studies are investigated on different shell type structures such as a cracked spherical shell and cracked cylindrical pipe with different crack orientations. The effects of using patch repairs with different fibre orientations on the reduction of stress intensity factors (SIFs) is also studied which can be useful for design purposes. XFEM is selected as any crack geometry can be embedded in the finite element mesh configuration with no need to coincide the crack geometry with meshed elements and so re‐meshing with fine mesh generation is not needed in the current method.     

Experimental and finite element analysis of fracture criterion in general yielding fracture mechanics

Efforts made over the last three decades to understand the fracture behaviour of structural materials in elastic and elasto-plastic fracture mechanics are numerous, whereas investigations related to fracture behaviour of materials in thin sheets or general yielding fracture regimes are limited in number. Engineering simulative tests are being used to characterize formability and drawability of sheet metals. However, these tests do not assure consistency in quality of sheet metal products. The prevention of failure in stressed structural components currently requires fracture mechanics based design parameters like critical load, critical crack-tip opening displacement or fracture toughness. The present attempt would aim to fulfill this gap and generate more information thereby increased understanding on fracture behaviour of sheet metals. In the present investigation, using a recently developed technique for determining fracture criteria in sheet metals, results are generated on critical CTOD and fracture toughness. Finite element analysis was performed to support the results on various fracture parameters. The differences are within 1 to 4%. At the end it is concluded that magnitude of critical CTOD and/or critical load can be used as a fracture criterion for thin sheets.