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 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.     

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.     

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.     

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.     

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.

     

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.     

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.     

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.     

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.     

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.     

Three-dimensional simulation of crack extension in brittle polycrystalline materials

Crack extension resistance in brittle polycrystals was investigated from the viewpoint of three-dimensional microcrack evolution. Even in the case of macroscopically two-dimensional cracks, inhomogeneous distribution of microscopic stress along the crack front gives rise to three-dimensional structures of extended crack surfaces. Numerical simulations of macroscopic crack extension were carried out, which showed that three-dimensional distribution of grain-by-grain thermal stress leads to a significant increase in the crack extension resistance. It was concluded that three-dimensional interpretation on the microscopic inhomogeneity is necessary for the correct comprehension of macroscopic crack extension behavior in brittle polycrystals.     

Numerical simulation of brittle fracture of thin-walled structures1

The model of brittle material, allowing for the determination of initiation and growth of cracks in thin-walled structures up to their destruction, is offered. Solutions of some problems are submitted: destruction of a plate at a homogeneous stress state; destruction of a beam at a pure bending; dynamic deformation and destruction of a plate with the concentrated mass having various initial speeds. Destruction of part of the material of the plate is shown to result in dynamic loss of stability of the solution.     

Parallel performance and scalability for block preconditioned finite element (p) solution of viscous flow

A two-dimensional p-type finite element scheme for distributed parallel computation of viscous flows is developed. The approach is based on an element-by-element implementation of the Biconjugate Gradient Stabilized 2 iterative method coupled with a recently developed class of block preconditioners. Critical to the overall parallel performance is the parallel solution of the imbedded bilinear preconditioner. Performance results are presented for the 2-D driven cavity incompressible viscous flow problem solved using incremental continuation in the Reynolds number on the Intel Touchstone Delta. These results are used to validate a run-time model. The run-time model is then used to examine the scaling properties of the method over a range of p and h.     

Combined atomistic simulation and continuum mechanics: Size-dependent behavior of atomistic simulation for brittle fracture in bcc-iron

The size-dependent behavior of atomistic simulation for brittle fracture in bcc-iron is studied using combined continuum-atomistic method. The result of displacement distribution at the crack tip indicates the radius of discrete region at the crack tip is about 120 Å, and the truncated distance of continuum elastic field is about 120 Å away from the crack tip. Further investigations of energy and atomic structure show that the elastic field of continuum mechanics cannot affect the crack tip processes effectively through the common region if the diameter of the atomistic region is smaller than 300 Å. We assume that the range of the system size that can bridge continuum with discrete region is at the scale of about 300 Å.     

Transient dynamic fracture analysis using scaled boundary finite element method: a frequency-domain approach

In the scaled boundary finite element method (SBFEM), the analytical nature of the solution in the radial direction allows accurate stress intensity factors (SIFs) to be determined directly from the definition, and hence no special crack-tip treatment, such as refining the crack-tip mesh or using singular elements (needed in the traditional finite element and boundary element methods), is necessary. In addition, anisotropic material behaviour may be handled with ease. These advantages are used in this study, in which a newly-developed Frobenius solution procedure in the frequency domain for solving the governing differential equations of the SBFEM, is applied to model transient dynamic fracture problems. The complex frequency-response functions are first computed using the Frobenius solution procedure. The dynamic stress intensity factors (DSIFs) are then extracted directly from the response functions. This is followed by a fast Fourier transform (FFT) of the transient load and a subsequent inverse FFT to obtain the time history of DSIFs. Benchmark problems with isotropic and anisotropic material behaviour are modelled using the developed frequency-domain approach. Excellent agreement is observed between the results of this study and those in published literature. The effects of the mesh density, the material internal damping coefficient, the maximum frequency and the frequency interval determining the frequency-response functions on the resultant accuracy and the computational cost are also discussed.