Mode I crack problems by fractal two level finite element methods

A semi-analytical method is suggested to determine the stress intensity factor (SIF) of two-dimensional (2D) crack problems. In it, the singularity is eliminated from the computational domain by the fractal two level finite element method (F2LFEM). In the present method, the fractal geometry concept and two level finite element method (2LFEM) are employed to automatically generate an infinitesimal mesh and transform these large number of degrees of freedom around the crack tip to a small set of generalized coordinates. By taking advantage of the same stiffness of 2D elements with similar shape, one transformation of the stiffness for the first layer of mesh is enough for all. This simple method is very economical in terms of computational time and computer memory. Highly accurate results of SIF and stresses are obtained.     

Body-force linear elastic stress intensity factor calculation using fractal two level finite element method

Fractal two level finite element method (F2LFEM) for the analysis of linear fracture problems subjected to body force loading is presented. The main objective here is to show that by employing the F2LFEM a highly accurate and an efficient linear analysis of fracture bodies subjected to internal loading can be obtained as it is hard to find any analytical and exact values of stress intensity factor (SIF) for any kind of geometry subjected to internal loading. Also in this paper, a fast method to transform the body force to the reduced force vector is presented and has been effectively employed. The problems solved here include both the single mode or mixed mode cracks subjected to internal body-force or external loading. In comparison with other numerical algorithms, it seems that with a small amount of computational time and computer storage, highly accurate results can be obtained.     

Modelling crack growth by level sets in the extended finite element method

An algorithm which couples the level set method (LSM) with the extended finite element method (X‐FEM) to model crack growth is described. The level set method is used to represent the crack location, including the location of crack tips. The extended finite element method is used to compute the stress and displacement fields necessary for determining the rate of crack growth. This combined method requires no remeshing as the crack progresses, making the algorithm very efficient. The combination of these methods has a tremendous potential for a wide range of applications. Numerical examples are presented to demonstrate the accuracy of the combined methods. Copyright © 2001 John Wiley & Sons, Ltd.     

Solution of two-dimensional elastic crack problems using a localized finite element method

Two-dimensional linear elastic fracture mechanics analysis of the opening-mode crack problem is carried out, in order to use a localized finite element method. The stress distribution near the crack-tip is stated in the form of eigensolutions obtained by a classical separation variables technique.     

Two-dimensional analysis of anisotropic crack problems using coupled meshless and fractal finite element method

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with fractal the finite element method (FFEM) for analyzing homogeneous, anisotropic, and two dimensional linear-elastic cracked structures subjected to mixed-mode (modes I and II) loading conditions. FFEM is adopted for discretization of domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprises both the element-free Galerkin and the finite element shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM-FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary and no post-processing (employing any path independent integrals) is needed to determine fracture parameters such as stress-intensity factors (SIFs) and T − stress. The numerical results based on all four orthotropic cases show that SIFs and T − stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions.     

Elastodynamic analysis of crack by finite element method using singular element

The finite element method using a singular element near the crack tip is extended to the elastodynamic problems of cracks where the displacement function of the singular element is taken from the solution of a propagating crack. The dynamic stress intensity factor for cracks of mode III or mode I deformations in a finite plate is determined.The results of computation for stationary cracks or propagating cracks under dynamic loadings are compared with the analytical solutions of other authors. It is shown that the present method satisfactorily describes the time variation of the stress intensity factor in dynamic crack problems.

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Résumé La méthode des éléments finis utilisant un élément singulier au voisinage de l'extrémité d'une fissure a été étendue aux problèmes élastodynamiques des fissures tels qu'ils se posent lorsque la fonction de déplacement d'un élément singulier est prise à partir de la solution d'une fissure en cours de propagation. Le facteur d'intensité des contraintes dynamiques correspondant à des fissures de mode III ou des déformations de mode I dans une plaque finie a été déterminé. Les résultats des calculs correspondant à des fissures stationnaires ou des fissures en cours de propagation sous des charges dynamiques sont comparées aux solutions analytiques obtenues par d'autres auteurs. On montre que la méthode présentée décrit de façon satisfaisante la variation en fonction du temps du facteur d'intensité des contraintes dans les problèmes de fissuration dynamique.

     

The fractal finite element method for unbounded problems

The fractal finite element method, previously developed for stress intensity factor calculation for crack problems in fracture mechanics, is extended to analyse some unbounded problems in half space. The concepts of geometrical similarity and two‐level finite element mesh are applied to generate an infinite number of self‐similar layers in the far field with a similarity ratio bigger than one; that is, one layer is bigger than the next in size but of the same shape. Only conventional finite elements are used and no new elements are generated. The global interpolating functions in the form of a truncated infinite series are employed to transform the infinite number of nodal variables to a small number of unknown coefficients associated with the global interpolating functions. Taking the advantage of geometrical similarity, transformation for one layer is enough because the relevant entries of the transformed matrix after assembling all layers are infinite geometric series of the similarity ratio and can be summed analytically. Accurate nodal displacements are obtained as shown in the numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.     

Coupled meshfree and fractal finite element method for mixed mode two‐dimensional crack problems

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 John Wiley & Sons, Ltd.     

A parametric study on the fractal finite element method for two‐dimensional crack problems

The Fractal Finite Element Method for calculating 2D stress intensity factors is modified by making the similarity ratio in the construction of the fractal mesh a variable. A parametric study is then carried out to examine the effects of the similarity ratio, the number of transformation terms, reduced integration and the initial crack opening angle on the quality of the numerical solutions. It is concluded that a large similarity ratio should be used to create the fractal mesh, and that reduced integration and a small initial crack opening angle may be used without producing significant errors in the solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

A two-dimensional augmented finite element for dynamic crack initiation and propagation

In this study, an implicit formulation of a 2D finite element based on a recently developed augmented finite element method is proposed for stable and efficient simulation of dynamic fracture in elastic solids. The 2D A-FE ensures smooth transition from a continuous state to a discontinuous state with an arbitrary intra-element cohesive crack, without the need of additional degree of freedoms (DoFs). Internal nodal DoFs are introduced for sub-domain integration and cohesive stress integration and they are then condensed at elemental level by a consistency-check based algorithm. The numerical performance of the proposed A-FE has been assessed through simulations of several benchmark dynamic fracture problems and in all cases the numerical results are in good agreement with the respective experimental results and other simulation results in literature. It has further been demonstrated that, (i) the dynamic A-FE is rather insensitive to mesh sizes and mesh structures; (ii) with similar solution accuracy it allows for the use of time steps 1–2 orders of magnitude larger than those used in other similar studies; and (iii) the implicit nature of the proposed A-FE allows for the use of a Courant number as large as 3.0–3.5 while maintaining solution stability.     

Extended finite element method for cohesive crack growth

The extended finite element method allows one to model displacement discontinuities which do not conform to interelement surfaces. This method is applied to modeling growth of arbitrary cohesive cracks. The growth of the cohesive zone is governed by requiring the stress intensity factors at the tip of the cohesive zone to vanish. This energetic approach avoids the evaluation of stresses at the mathematical tip of the crack. The effectiveness of the proposed approach is demonstrated by simulations of cohesive crack growth in concrete.     

Numerical study of dynamic crack growth by the finite element method

Recent developments in numerical techniques for dynamic transient stress analysis have ensured that realistic models can now be employed in crack propagation studies. In this paper transient dynamic finite element solutions are undertaken for both double cantilever beam (DCB) and pipeline problems with propagation of the crack being permitted. Standard parabolic isoparametric elements are employed for spatial discretization with an explicit (central difference) scheme being employed for time integration. Both critical stress and energy balance crack propagation criteria are considered.The pressurised pipeline problem is solved for as a fully three-dimensional solid. Firstly, a stationary crack is considered and both large deformations and plasticity effects are accounted for. The transient case of a dynamically propagating crack is then modelled, employing both a stress and energy criterion. Elastic large deformation behaviour is permitted for this case.

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Résumé Des développements récents dans les techniques numériques pour l'analyse des contraintes dynamiques transitoires ont permis d'utiliser à présent des modèles réalistes dans les études de propagation des fissures. Dans ce mémoire, on envisage des solutions par éléments finis pour les transitoires dynamiques dans les cas de la poutre double cantilever et de problèmes de pipelines où l'on autorise la propagation d'une fissure. On recourt aux éléments paramétriques paraboliques standards pour réaliser une division discrète de l'espace, et l'on utilise pour l'intégration dans le temps un schéma explicite à différence centrale. On considére à la fois les critères de contraintes critiques et d'équilibre d'énergie lors de la propagation de la fissure. Le problème du pipeline pressurisé est solutionné en considérant ce dernier comme un solide tridimensionnel. En premier lieu, on considère une fissure stationnaire et l'on tient compte des effets des grandes déformations et de la plasticité. On met ensuite en équation le cas transitoire d'une fissure en propagation dynamique, en utilisant un critère de contrainte et un critère d'énergie. Ce cas permit d'envisager le comportement sous des déformations élastiques importantes.

     

Modeling of dynamic crack branching by enhanced extended finite element method

The conventional extended finite element method (XFEM) is enhanced in this paper to simulate dynamic crack branching, which is a top challenge issue in fracture mechanics and finite element method. XFEM uses the enriched shape functions with special characteristics to represent the discontinuity in computation field. In order to describe branched cracks, it is necessary to set up the additional enrichment. Here we have developed two kinds of branched elements, namely the “element crossed by two separated cracks” and “element embedded by a junction”. Another series of enriched degrees of freedom are introduced to seize the additional discontinuity in the elements. A shifted enrichment scheme is used to avoid the treatment of blending element. Correspondingly a new mass lumping method is developed for the branched elements based on the kinetic conservation. The derivation of the mass matrix of a four-node quadrilateral element which contains two strong discontinuities is specially presented. Then by choosing crack speed as the branching criterion, the branching process of a single mode I crack is simulated. The results including the branching angle and propagation routes are compared with that obtained by the conventionally used element deletion method.     

Transient two-dimensional heat conduction problems solved by the finite element method

A finite element weighted residual process has been used to solve transient linear and non-linear two-dimensional heat conduction problems. Rectangular prisms in a space-time domain were used as the finite elements. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in tables with analytical, as well as other numerical data. The finite element method compared favourably with these results. It was found to be stable, convergent to the exact solution, easily programmed, and computationally fast. Finally, the method does not require constant parameters over the entire solution domain.     

Three-dimensional mixed mode analysis of a cracked body by fractal finite element method

A semi-analytical method namely fractal finite element method is presented for the determination of stress intensity factor for the straight three-dimenisonal plane crack. Using the concept of fractal geometry, infinite many of finite elements are generated virtually around the crack border. Based on the analytical global displacement function, numerous DOFs are transformed to a small set of generalised coordinates in an expeditious way. No post-processing and special finite elements are required to develop for extracting the stress intensity factors. Examples are given to illustrate the accuracy and efficiency of the present method. Very good accuracy (with less than 3% errors) is obtained for the maximum value of SIFs for different modes.     

A finite element method for crack growth without remeshing

An improvement of a new technique for modelling cracks in the finite element framework is presented. A standard displacement‐based approximation is enriched near a crack by incorporating both discontinuous fields and the near tip asymptotic fields through a partition of unity method. A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed. This technique allows the entire crack to be represented independently of the mesh, and so remeshing is not necessary to model crack growth. Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique. Copyright © 1999 John Wiley & Sons, Ltd.     

Mixed-mode delamination growth of laminar composites by using three-dimensional finite element modeling

ABSTRACT Delamination is one of the most frequent failure modes in laminated composites. Its importance is crucial, because a delamination can occur in the interior of a panel without any noticeable damage on the surface, drastically reducing its strength and stiffness. A study has to be made on critical dimensions of delaminations and their shape, through the calculation of the strain energy release rate (SERR), G. This study was performed numerically, for a given geometry, with varying loads and shapes of delamination, in pure and mixed‐mode propagation. All numerical values were obtained with three‐dimensional finite element (FE) analyses from a commercial package. The use of three‐dimensional analyses in simple geometries helps establish the basis for the more complex ones, and the correspondence with the usual analytical or numerical bi‐dimensional plane‐strain analysis. The conclusions were (a) G is not constant along the crack tip, even for mode I propagation and straight crack tip; (b) the mean value of G obtained from a three‐dimensional analysis equals the value obtained in bi‐dimensional plane‐strain analysis; (c) in mixed‐mode propagation, the method exhibits a good correlation with experimental results and (d) the shape and mode partitioning of the SERR depend not only on the loading, but also on the shape of the crack front.     

Influence of minimum element size to determine crack closure stress by the finite element method

Measuring opening or closure stress is a complex process that influences the low accuracy of obtained data. Finite element models have been one of the available ways to deal with this problem. The difficulty of modelling the whole process of crack growth (due to the great number of cycles implied) as the great complexity of the phenomenon itself (with a high plastic strain concentrated in a small area, with elevated stress gradients) has made the results to be quite varied, being influenced by a great number of modelling parameters. Of those parameters, the minimum size of the element used to mesh the area around the crack tip vicinity presents a great influence on the results.In this work, a detailed analysis of the influence of this parameter in the results in terms of closure or opening stress is presented. The effect that different meshing criteria can have on the result is complex and it has been necessary to reduce the element size around the crack tip to a size that had not been reached before. Procedures and modelling criteria stricter than the ones shown in the current bibliography are proposed. A methodology for the correct interpretation of the results is also established.