Reappraisal of the quarter-point quadrilateral element in linear elastic fracture mechanics

The eight noded quarter-point Serendipity quadrilateral isoparametric element is reexamined. The stresses are proven to be square-root singular on all rays in a small region adjacent to the crack tip and, as was previously shown, along the element sides. It is demonstrated that the element strain energy, and hence its stiffness, is bounded. The effect of element size in characterizing the square-root singular behavior is investigated through stress intensity factor calculations in the case of two geometries with crack tip elements of various dimensions. Workers in the field of fracture mechanics may now, without hesitation, employ this element for modeling crack tip singularities in linear elastic material.

---

Résumé On réexamine la validité d'un élément quadrilatère isoparamétrique à huit noeuds quart-point de Serendipity. On prouve que les contraintes présentent une singularité du deuxième ordre sur tous les rayons situés dans une petite région adjacente à l'extrémité de la fissure et, comme on l'a montré précédemment, le long des bords de l'élément. On démontre que l'énergie de déformation de l'élément, et donc sa rigidité, est limitée. On étudie l'effet de la taille de l'élément sur la caractérisation de la singularité du deuxième ordre, en calculant le facteur d'intensité d'entaille dans le cas de deux géométries présentant des éléments de dimensions différentes à l'extrémité d'une fissure. Les spécialistes en mécanique de la rupture peuvent à présent, sans hésitation, utiliser cet élément pour modéliser les singularités qui se présentent à l'extrémité d'une fissure dans un matériau linéaire et élastique.

     

An explicit residual‐type error estimator for ‐quadrilateral extended finite element method in two‐dimensional linear elastic fracture mechanics

This paper deals with explicit residual a posteriori error estimation analysis for ‐quadrilateral extended finite element method (XFEM) discretizations applied to the two‐dimensional problem of linear elastic fracture mechanics. The result is twofold. First, to enable estimation procedures with application to XFEM, a specific quasi‐interpolation operator of averaging type is constructed. The main challenge here arises from the different types of enrichments implemented, and hence, to impose the constant‐preserving property of the interpolation operator on an element, we use the idea of an extension operator. An upper bound on the discretization error measured in the energy norm and associated local error indicators are then constructed and analyzed. The second result follows from the error analysis and concerns an alternative choice of branch functions used in XFEM applications. In particular, the branch functions have to be chosen to fulfill the divergence‐free conditions within the crack tip element and traction‐free boundary conditions on the crack faces. Then, the corresponding XFEM solution gains a better accuracy with less degrees of freedom. Finally, numerical examples are provided with comparative results. Copyright © 2012 John Wiley & Sons, Ltd.     

Crack stability in linear elastic fracture mechanics

A linear elastic fracture mechanics analysis of the conditions that produce crack instability is presented. If the initial crack extension causes the change in crack-tip resistance to be negative with respect to the change in crack-tip loading, the crack will continue to propagate even though the loading agent remains stationary, and the crack is defined as unstable. The value of crack-tip load when the crack becomes unstable, G _c, is not only a function of the plate material and thickness and fracture mode, but also depends on the specimen geometry and size, and on the compliance of the loading system. The crack-tip resistance, G _R, on the other hand, is essentially a property of the plate material and thickness and fracture mode if the crack-propagation is time independent. Once G _R has been experimentally determined as a function of crack-propagation distance for a particular plate material and thickness and fracture mode, the value of G _e can be calculated for the same material and thickness and fracture mode for any plate configuration for which the elastic stress analysis is known.

---

Zusammenfassung In dem Gebiet der Mechanik elastischer linearer Risse ist eine Analyse der Bedingungen, die zur Sprungbestdndigkeit führen, dargestellt. Wenn die ursprüngliche Sprungverlängerung veranlaßt, daß sick die Veranderung in dem Widerstand des Sprungendes in Bezug auf die Veranderung in der Belastung des Sprungendes negativ verhält, wird der Sprung sich weiter verlängern, obwohl das Belastungsmittel stillsteht, and der Sprung wird als widerstandsunfähig bezeichnet. Der Wert der Sprungendenbelastung wenn der Sprung widerstandsunfäbig wird, G _c, ist nicht nur eine Funktion des Plattenmaterials, des Plattendicke and dert Art des Risses, sondern hängt auch von der Geometric und der Größe des Stückes, Bowie von der Anpassung des Belastungssystems ab. Wenn, andererseits, die Sprungvergrößerung unabhängig von der Zeit vor sich geht, so ist die Sprungendenfestigkeit, G _R, im wesentlichen die Eigenschaft des Plattenmaterials, des Plattendicke and der Art des Risses. Wenn G _R erst einmal durch Versuche als Funktion der Länge der Sprungvergrößerung für ein bestimmtes Plattenmaterial, Plattendicke and RiBart bestimmt ist, so kann der Wert von G _c für das selbe Material, Dicke and Rißart in jeder Plattengeometrie, für welche die Analyse der elastischen Beanspruchung bekannt ist, bezechnet werden.

---

Résumé On rapporte l'analyse des conditions produisant l'instabilité des fissures. Au cas où le prolongement de la fissure initiale modifie la résistance de la pointe de la criqûre à une valeur négative, au regard du changement en charge de la pointe, la fissure continuera à se propager même quand le système de charge reste stationnaire. Dans ce cas, la fissure est considéré comme instable. La valeur de la charge de la pointe d'une fissure instable (G _c) n'est pas seulement une caractéristique du matériau du panneau mais dépend en plus de la géometrie de l'échantillon et de ses dimensions ainsi que de la complaisance du système de charge. D'autre part, la résistance de la pointe de la fissure (G _R) est essentiellement une caractéristique du matériau du panneau et de son épaisseur ainsi que de son mode de rupture si la propagation de la criqure est indépendant du temps. Autant que l'on aura déterminé G _R, par l'expérience, en fonction de la distance de propagation de la fissure pour un matériau de panneau donné et de son épaisseur ainsi que de son mode de rupture, la valcur de G _c pent être calculé, pour un même matériau de même épaisseur et mode de rupture, pour toute configuration en panneau dont on sait l'analyse élastique.

     

Linear elastic fracture mechanics applied to cracked plates and shells

A general theory concerning through-cracks in plates and shells is proposed. Applying the method of local development of the kinematic unknowns to a shell of arbitrary shape, the distribution of displacements and rotations as functions of polar coordinates in the vicinity of the crack tip is given and the five corresponding intensity factors are defined. New path-independent integrals are introduced and related to the intensity factors so that these can be evaluated numerically. Finally, a fracture criterion for plane problems is extended to shells.

---

Résumé On propose une théorie générale du comportement des fissures traversantes dans les plaques et les coques. Pour cela, on applique la méthode du développement limité des variables cinématiques à une coque de forme quelconque. On obtient la distribution des déplacements et des rotations en fonction des coordonnées polaires dans le voisinage de la singularité et on définit les cinq facteurs d'intensité correspondants. On propose de nouvelles intégrales curvilignes independantes du contour et on les relie aux facteurs d'intensité de telle sorte que ces derniers peuvent être calculés numériquement. Enfin on étend aux coques un critère de rupture utilisé jusqu'alors dans les problèmes plans.

     

On the relevance of linear elastic fracture mechanics to ice

A new criterion is proposed which allows one to estimate the minimum size for a brittle ice fracture specimen comprised of a small number of grains. According to this criterion, linear elastic fracture mechanics is a useful theory for fresh-water ice but may have limited use for saline ice.     

An assumed displacement hybrid finite element model for linear fracture mechanics

This paper deals with a procedure to calculate the elastic stress intensity factors for arbitrary-shaped cracks in plane stress and plane strain problems. An assumed displacement hybrid finite element model is employed wherein the unknowns in the final algebraic system of equations are the nodal displacements and the elastic stress intensity factors. Special elements, which contain proper singular displacement and stress fields, are used in a fixed region near the crack tip; and the interelement displacement compatibility is satisfied through the use of a Lagrangean multiplier technique. Numerical examples presented include: central as well as edge cracks in tension plates and a quarter-circular crack in a tension plate. Excellent correlations were obtained with available solutions in all the cases. A discussion on the convergence of the present solution is also included.     

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.     

Compatibility of linear elastic (k1c) and general yielding (COD) fracture mechanics

An investigation of the applicability of the general yielding fracture mechanics concept of crack opening displacement is made in relation to the well established concepts of linear elastic fracture mechanics. The nature of the relationship between crack opening displacement and stress intensity factor is explored using the concepts of linear elasticity and model analyses proposed by Dugdale. Tests on C/Mn steel, a low alloy steel and a manual metal arc weld deposit define the nature of this relationship in the stress region for which the theoretical compatibility is no longer to be expected. Based on this relationship a single fracture test procedure may be used to determine COD or K_1c (depending on the stress conditions at failure). The results of the tests analysed in terms of J, the contour integral, determine the significance of J as a fracture characterising parameter. Defect tolerance parameters for the assessment of the significance of flaws in welded structures are also introduced.     

New path independent integrals in linear elastic fracture mechanics

In this paper the properties of eigenfunction expansion form (abbreviated as EEF) in the crack problems of plane elasticity and antiplane elasticity are discussed in details. After using the Betti's reciprocal theorem to the cracked body, several new path independent integrals are obtained. All the coefficients in the EEF at the crack-tip, including the K₁, K₂ and k₃ values, can be related to corresponding path-independent integrals.     

The perturbation method and the extended finite element method. An application to fracture mechanics problems

The extended finite element method has been successful in the numerical simulation of fracture mechanics problems. With this methodology, different to the conventional finite element method, discretization of the domain with a mesh adapted to the geometry of the discontinuity is not required. On the other hand, in traditional fracture mechanics all variables have been considered to be deterministic (uniquely defined by a given numerical value). However, the uncertainty associated with these variables (external loads, geometry and material properties, among others) it is well known. This paper presents a novel application of the perturbation method along with the extended finite element method to treat these uncertainties. The methodology has been implemented in a commercial software and results are compared with those obtained by means of a Monte Carlo simulation.     

A reappraisal of transition elements in linear elastic fracture mechanics

This article offers a detailed comparison of the transition elements described by P.P. Lynn and A.R. Ingraffea [International Journal for Numerical Methods in Engineering 12,1031–1036] and C. Manu[Engineering Fracture Mechanics 24,509–512]. The source of a numerical phenomenon in using Manu's transitionelement (TE) is explained. The effect of eight-noded TEs with differentquarter-point elements (QPE) on the calculated stress intensity factors (SIFs) isinvestigated. Strain at the crack tip is shown to be singular for any ray emanating from the crack tip within an eight-noded TE, but strain has bothr ^–1/2andr ^–1singularities, withr ^–1/2dominating for large TEs. Semi-transition elements (STEs) are defined and shown to have a marginal effect on the calculated SIFs. Nine-nodedtransition elements are formulated whose strain singularity is shown to be the same as that of eight-noded TEs. Then the effect of eight-noded and nine-noded TEs with collapsed triangular QPEs, and rectangular and nonrectangular quadrilateral eight-noded and nine-noded QPEs, is studied, and nine-noded TEs are shown to behave exactly like eight-noded TEs with rectangular eight-noded and nine-noded QPEs and to behave almost the same with other QPEs. The layered transition elements proposed by V. Murti and S.Valliapan [Engineering Fracture Mechanics 25, 237–258] areformulated correctly. The effect of layered transition elements is shown by two numerical examples.     

An extended finite element method applied to levitated droplet problems

We consider a standard model for incompressible two‐phase flows in which a localized force at the interface describes the effect of surface tension. If a level set method is applied then the approximation of the interface is in general not aligned with the triangulation. This causes severe difficulties w.r.t. the discretization and often results in large spurious velocities. In this paper we reconsider a (modified) extended finite element method (XFEM), which in previous papers has been investigated for relatively simple two‐phase flow model problems, and apply it to a physically realistic levitated droplet problem. The results show that due to the extension of the standard FE space one obtains much better results in particular for large interface tension coefficients. Furthermore, a certain cut‐off technique results in better efficiency without sacrificing accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Minimum theorems in 3D incremental linear elastic fracture mechanics

The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen in Appl Mech Rev 47, 1994, Stability and nonlinear solid mechanics. Wiley, New York, 2000; Mielke in Handbook of differential equations, evolutionary equations. Elsevier, Amsterdam, 2005; Bourdin et al. in The variational approach to fracture. Springer, Berlin, 2008). In a recent publication (Salvadori and Carini in Int J Solids Struct 48:1362–1369, 2011) minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. They were reminiscent of Ceradini’s theorem (Ceradini in Rendiconti Istituto Lombardo di Scienze e Lettere A99, 1965, Meccanica 1:77–82, 1966) in plasticity. Following the cornerstone work of Rice (1989) on weight function theories, Leblond et al. (Leblond in Int J Solids Struct 36:79–103, 1999; Leblond et al. in Int J Solids Struct 36:105–142, 1999) proposed asymptotic expansions for stress intensity factors in three dimensions—see also Lazarus (J Mech Phys Solids 59:121–144, 2011). As formerly in 2D, expansions can be given a Colonnetti’s decomposition (Colonnetti in Rend Accad Lincei 5, 1918, Quart Appl Math 7:353–362, 1950) interpretation. In view of the expression of the expansions proposed in Leblond (Int J Solids Struct 36:79–103, 1999), Leblond et al. (Int J Solids Struct 36:105–142, 1999) however, symmetry of Ceradini’s theorem operators was not evident and the extension of outcomes proposed in Salvadori and Carini (Int J Solids Struct 48:1362–1369, 2011) not straightforward. Following a different path of reasoning, minimum theorems have been finally derived.     

The use of linear elastic fracture mechanics for particulate solids

The fracture of model porous particulate solids consisting of sand and low volume fractions of a polymeric binder has been investigated. In order to apply a linear elastic fracture mechanics analysis it was necessary to use an effective crack length comprising the sum of the notch depth and a quantity a. The behaviour of the model solids was intermediate between that exhibited by ceramics and polymers. A possible interpretation of a is that it represents a process zone size where the energy dissipation mechanism could involve either microcracking or yielding of the interparticle junctions in the zone. The former would correspond to ceramic-like behaviour while the latter is characteristic of polymers.     

Limitations of elastic definitions in Al-Si-Mg cast alloys with enhanced plasticity: linear elastic fracture mechanics versus elastic-plastic fracture mechanics

Linear elastic fracture mechanics describes the fracture behavior of materials and components that respond elastically under loading. This approach is valuable and accurate for the continuum analysis of crack growth in brittle and high strength materials; however it introduces increasing inaccuracies for low-strength/high-ductility alloys (particularly low-carbon steels and light metal alloys). In the case of ductile alloys, different degrees of plastic deformation precede and accompany crack initiation and propagation, and a non-linear ductile fracture mechanics approach better characterizes the fatigue and fracture behavior under elastic-plastic conditions.To delineate plasticity effects in upper Region II and Region III of crack growth an analysis comparing linear elastic stress intensity factor ranges (ΔK_el) with crack tip plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) is presented. To compute plasticity corrected stress intensity factor ranges (ΔK_pl), a new relationship for plastic zone size determination was developed taking into account effects of plane-strain and plane-stress conditions (“combo plastic zone”). In addition, for the upper part of the fatigue crack growth curve, elastic-plastic (cyclic J based) stress intensity factor ranges (ΔK_J) were computed from load-displacement records and compared to plasticity corrected stress intensity factor ranges (ΔK_pl). A new cyclic J analysis was designed to compute elastic-plastic stress intensity factor ranges (ΔK_J) by determining cumulative plastic damage from load-displacement records captured in load-control (K-control) fatigue crack growth tests. The cyclic J analysis provides the true fatigue crack growth behavior of the material. A methodology to evaluate the lower and upper bound fracture toughness of the material (J_IC and J_max) directly from fatigue crack growth test data (ΔK_FT(JIC) and ΔK_FT(Jmax)) was developed and validated using static fracture toughness test results. The value of ΔK_FT(JIC) (and implicitly J_IC) is determined by comparing the plasticity corrected elastic fatigue crack growth curve with the elastic-plastic fatigue crack growth curve. A most relevant finding is that plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) are in remarkably good agreement with cyclic J analysis results (ΔK_J), and provide accurate plasticity corrections up to a ΔK corresponding to J_IC (i.e. ΔK_FT(JIC)). Towards the end of the fatigue crack growth test (above ΔK_FT(JIC)) when plasticity is accompanied by significant tearing, the cyclic J analysis provides a more accurate way to capture the true behavior of the material and determine ΔK_FT(Jmax). A procedure to decouple and partition plasticity and tearing effects on crack growth rates is given.Three cast Al-Si-Mg alloys with different levels of ductility, provided by different Si contents and heat treatments (T61 and T4) are evaluated, and the effects of crack tip plasticity on fatigue crack growth are assessed. Fatigue crack growth tests were conducted at a constant stress ratio, R = 0.1, using compact tension specimens.     

On quarter-point three-dimensional finite elements in linear elastic fracture mechanics

Use of the finite element method for calculating stress intensity factors of two-dimensional cracked bodies has become commonplace. In this study, the more difficult task of applying finite elements to three-dimensional cracked bodies is investigated. Since linear elastic material is considered, square root singular stresses exist along the edge of an embedded crack. To deal with this numerical difficulty, twenty noded, isoparametric, serendipity, quarter-point, singular, solid elements are employed. Examination of these elements is carried out in order to determine the extent of the singular behavior.In addition, the stiffness derivative technique is explored, together with quarter-point elements, to determine an accurate procedure for computing stress intensity factors in three-dimensions. The problem of chosing a proper virtual crack extension is addressed. To this end, the disturbance in the square root singular stresses is examined and compared with a similar disturbance which occurs in two-dimensions. As a numerical example, a pennyshaped crack in a finite height cylinder is considered with various meshes. It is found that stress intensity factors can be calculated to an accuracy within 1 percent when quarter-point cylindrical elements are employed with the stiffness derivative technique such that the crack extension is one in which one corner node is not moved, the other corner node is moved a small distance, and the midside node is moved one-half that distance. This crack extension is analogous to that of a straight crack advance for a brick element. Both of these crack advances disturb the square root singular stresses in a manner similar to that which occurs with the two-dimensional eight noded element in which the crack has been advanced a small distance.     

An optimally convergent discontinuous Galerkin‐based extended finite element method for fracture mechanics

The extended finite element method (XFEM) enables the representation of cracks in arbitrary locations of a mesh. We introduce here a variant of the XFEM rendering an optimally convergent scheme. Its distinguishing features are as follows: (a) the introduction of singular asymptotic crack tip fields with support on only a small region around the crack tip (the enrichment region), (b) only one and two enrichment functions are added for anti‐plane shear and planar problems, respectively and (c) the relaxation of the continuity between the enrichment region and the rest of the domain, and the adoption of a discontinuous Galerkin (DG) method therein. The method is provably stable for any positive value of a stabilization parameter, and by weakly enforcing the continuity between the two regions it eliminates ‘blending elements’ partly responsible for the suboptimal convergence of some early XFEMs. Moreover, the particular choice of enrichment functions results in a surprisingly sparse stiffness matrix that remains reasonably conditioned as the mesh is refined. More importantly, the stress intensity factors can be extracted with a satisfactory accuracy as primary unknowns. Quadrature strategies required for the optimal convergence are also discussed. Finally, the DG method was modified to retain stability based on an inf‐sup condition. Copyright © 2009 John Wiley & Sons, Ltd.