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.

     

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.     

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.     

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.     

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.     

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.     

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.     

Automated simulation of fatigue crack propagation for two-dimensional linear elastic fracture mechanics problems by boundary element method

In this paper, automated simulation of multiple crack fatigue propagation for two-dimensional (2D) linear elastic fracture mechanics (LEFM) problems is developed by using boundary element method (BEM). The boundary element method is the displacement discontinuity method with crack-tip elements proposed by the author. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Local discretization on the incremental crack extension is performed easily. Further the new adding elements and the existing elements on the existing boundaries are employed to construct easily the total structural mesh representation. Here, the mixed-mode stress intensity factors are calculated by using the formulas based on the displacement fields around crack tip. The maximum circumferential stress theory is used to predict crack stability and direction of propagation at each step. The well-known Paris’ equation is extended to multiple crack case under mixed-mode loadings. Also, the user does not need to provide a desired crack length increment at the beginning of each simulation. The numerical examples are included to illustrate the validation of the numerical approach for fatigue growth simulation of multiple cracks for 2D LEFM problems.     

Crack tip displacement factor-D: an application to two dimensional linear elastic fracture mechanics

A fracture parameter, crack tip displacement factor D is introduced to characterize the two-dimensional linear elastic fracture mechanics problems. Similar to the conventional COD approach, the D factor can be determined directly from the crack surface displacement. The major advantage of using this fracture control parameter is that one can obtain it by evaluating the crack surface displacement from a numerical analysis or an experiment without the need for considering the stress field at the crack tip. Unlike the COD method, the introduced fracture parameter may be used for not only the simple opening mode but also for the sliding and tearing modes as well as for the mixed mode fracture problems. This article presents a detailed derivation of the D factor as well as some sample applications.     

On the applicability of linear elastic fracture mechanics to environmental stress cracking of low-density polyethylene

The primary aim of this work was to re-examine the range of applicability of linear elastic fracture mechanics (LEFM) to environmental stress cracking of low-density polyethylene. By use of a number of specimen types and a range of specimen dimensions and loads it is shown that theK _I-c relationship is unique only at lowK _Is. The material studied showed a region of constant crack velocity which was not caused by plasticity effects or the failure of LEFM. However, crack arrest, which occurred at highK _Is or loads, was shown to be caused by ductile yielding causing the crack to blunt.     

A boundary element method for two dimensional linear elastic fracture analysis

A boundary element method is developed for the analysis of fractures in two-dimensional solids. The solids are assumed to be linearly elastic and isotropic, and both bounded and unbounded domains are treated. The development of the boundary integral equations exploits (as usual) Somigliana's identity, but a special manipulation is carried out to regularize certain integrals associated with the crack line. The resulting integral equations consist of the conventional ordinary boundary terms and two additional terms which can be identified as a distribution of concentrated forces and a distribution of dislocations along each crack line. The strategy for establishing the integral equations is first outlined in terms of real variables, after which complex variable techniques are adopted for the detailed development. In the numerical implementation of the formulation, the ordinary boundary integrals are treated with standard boundary element techniques, while a novel numerical procedure is developed to treat the crack line integrals. The resulting numerical procedure is used to solve several sample problems for both embedded and surface-breaking cracks, and it is shown that the technique is both accurate and efficient. The utility of the method for simulating curvilinear crack propagation is also demonstrated.     

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.     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.     

Surface integral finite element hybrid (SIFEH) method for fracture mechanics

An effective surface integral and finite element hybrid (SIFEH) method has been developed to model fracture problems in finite plane domains. This hybridization by (incrementally) linear superposition combines the best features of both component methods. Finite elements are used to model the finite domain (and eventually nonlinearity), while continuous distributions of dislocations (resulting in surface integral equations) are used to model the fracture (i.e. displacement discontinuity). This method has been implemented in a computer program and results of representative problems are presented: these compare very well with known solutions and they demonstrate the computational advantages of SIFEH over other numerical methods (including the individual components).