The optimum size of quarter-point crack tip elements

Ingraffea and Manu¹ and Lynn and Ingraffea² have shown that the size of the quarter-point elements can affect the computed elastic stress intensity factor. The nature of the effect is such that, all other details remaining constant, there is a particular crack tip element size which minimizes the error in the computed stress intensity factor. Here, size of element means the radial edge length. The reasons for this size dependence are discussed below. It will be seen that the discussion is in terms of the need to simultaneously represent the singular and finite stress terms in a given problem. The discussion has relevance to other formulations of crack tip elements.     

Improved quarter-point crack tip element

We present a modification to the quarter-point crack tip element and employ this element in two-dimensional boundary integral fracture analysis. The standard singular element is adjusted so that the near-tip crack opening displacement satisfies a known constraint: the coefficient of the term which is linear in the distance to the tip must vanish. Stress intensity factors calculated with the displacement correlation technique are shown to be highly accurate, and significantly more accurate than with the standard element. The improvements are especially dramatic for mixed-mode problems involving curved and interacting cracks.     

Hypersingular quarter-point boundary elements for crack problems

The present paper deals with the study and effective implementation for Stress Intensity Factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation. Expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented. The integration is carried out by transformation of the hypersingular integrals into regular integrals, which are evaluated by standard quadratures, and simple singular integrals, which are integrated analytically. The generality of the method allows for the modelling of curved cracks and the use of straight line quarter-point elements. The Stress Intensity Factors can be computed very accurately from the Crack Opening Displacement at collocation points extremely close to the crack tip. Several examples with different crack geometries are analyzed. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip.     

Finite elements for determination of crack tip elastic stress intensity factors

A new type of finite element is introduced which embodies the inverse square root singularity present near a crack in an elastic medium. Using this element near the tip in two typical cracked configurations, stress intensity factors within 5 per cent of accepted values were obtained with meshes having as few as 250° of freedom.     

Unsteady growth of mode III crack in elastic perfectly-plastic material

In this paper the assembly of the near-tip fields given by J. R. Rice is completed for the mode III crack growing quasi-statically and unsteadily in elastic perfectly-plastic material. The obtained results provide a particular example for the general theoretical relations between the steady state and unsteady state crack growth. Further, the general expression of the rate of crack opening displacement is obtained, which is similar to one by J.R. Rice and co-workers for mode I crack growing in elastic perfectly-plastic material. The fracture criterion of the critical opening displacement at a prescribed distance behind the crack tip is discussed. As a result, the theoretical J-resistance curves are given.     

On the use of special crack tip elements in cracked elastic sheets

This paper presents a finite element method for determining the stress intensity factor in a cracked elastic sheet. Special cracked elements are placed around each crack tip; in each special element the stresses and displacements are derived from the exact stress function while the continuity of the displacements at the nodes is satisfied in a least square sense. A general procedure for evaluating the stiffness matrix of a cracked isotropic, or orthotropic, element is presented, and the numerical results obtained are compared with exact analytical results.

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Résumé Le mémoire présente une méthode par éléments finis pour déterminer le facteur d'intensité de contrainte dans un feuillard élastique fissuré. Des éléments spéciaux, présentant des fissures, sont disposés autour de chaque extrémité de fissures. Dans chacun de ces éléments spéciaux, les contraintes et les déplacements sont déduits de la fonction exacte d'une contrainte tandis que la continuité des déplacements aux noeuds satisfait à une règle des moindres carrés. Une procédure générale pour l'évaluation de la matrice de rigidité d'un élément fissuré à caractère isotrope ou orthotrope est présentée, et les résultats numériques obtenus sont comparés avec les résultats analytiques exacts.

     

Nonlocal elastic damage near crack tip

A constitutive modeling for nonlocal elastic damage near crack tip is proposed. A calculation method for nonlocal elastic damage is introduced and the computational results for stress and damage are given by means of finite element method.     

Plane stress crack-line fields for crack growth in an elastic perfectly-plastic material

Mode-I crack growth in an elastic perfectly-plastic material under conditions of generalized plane stress has been investigated. In the plastic loading zone, near the plane of the crack, the stresses and strains have been expanded in powers of the distance, $\text{y}$, to the crack line. Substitution of the expansions in the equilibrium equations, the yield condition and the constitutive equations yields a system of simple ordinary differential equations for the coefficients of the expansions. This system is solvable if it is assumed that the cleavage stress is uniform on the crack line. By matching the relevant stress components and particle velocities to the dominant terms of appropriate elastic fields at the elastic-plastic boundary, a complete solution has been obtained for ?_y in the plane of the crack. The solution depends on crack-line position and time, and applies from the propagating crack tip up to the moving elastic-plastic boundary. Numerical results are presented for the edge crack geometry.     

Reconsiderations on singularity or crack tip elements

Various singularity finite elements proposed so far are reviewed and their correlations are discussed. The method adopted in the comparative study is based on the singularity mapping or transformation technique. It is suggested that a compact design of computational routines is materialized by using the element which embodies singularity transformation and can be compatible with the notion of using the variable-number-nodes element in the general-purpose computer programs.     

Dynamic effects on the near crack-line fields for crack growth in an elastic perfectly-plastic solid

The dynamic effects on the near crack-line fields for steady-state tensile crack growth in an elastic perfectly-plastic solid are investigated under plane stress condition in this paper. In the plastic loading zone, the stresses and particle velocities near the crack-line are expanded in powers of the distance y to the crack line, with coefficients which depend on the distance of the moving crack tip. Substituting the expansions into the equations of motion, the Huber-Mises yield criterion and the Prandtl-Reuss flow rule yield a system of non-linear ordinary differential equations for the coefficients. This equation system is solved by using the approximate approach proposed by J.D. Achenbach and Z.L. Li. Finally, the crack growth criterion of critical strain is employed to determine the value of the remote elastic stress intensity factor K ₁that would be required for a crack growing steadily at a given Mach number. It is also shown in this paper that the steady-state dynamic solution yields the quasi-static solution as the speed of crack growth tends to zero.

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Résumé On étude les effets dynamiques qu'exercent les champs de contrainte au voisinage de la ligne de fissuration sous des conditions de tension stable dans un solide parfaitement élastique-plastique et pour un état plan de tension.Dans la zone de sollicitation plastique, les contraintes et vitesses élémentaires au voisinage de la ligne de fissuration se distribuent selon une puissance de la distance y à partir de cette ligne, avec des coefficients qui sont eux-mêmes dépendant de la distance à l'extrémité de la fissure en mouvement. En substituant les dilatations dans les équations de mouvement, le critère de plastification de Huber-Von Mises et la loi de'ecoulement de Prandtl-Reuss conduisent à un systéme d'équations différentielles ordinaires non linéaires pour déterminer ces coefficients.On résoud ce système d'équations grâce à une approach proposée par J.D. Achenbach et Z.L. Li.Enfin, le critère de déformation critique entraînant une croissance de la fissure est utilisé pour déterminer la valeur du facteur d'intensité des contraintes élastiques lointaines K ₁qui serait requis pour qu'une fissure croisse régulièrement à un nombre de Mach donné. On montre également dans l'étude qu'une solution dynamique sous conditions stables conduit à une solution quasi-statique lorsque la vitesse de propagation de la fissure tend vers zéro.

     

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid are investigated under plane strain and plane stress conditions. In each case, by expanding the plastic fields and the governing equations in the coordinate y, the problem is reduced to solving a system of nonlinear ordinary differential equations which is similar to that of mode III derived by Achenbach and Z.L.Li. An approximate solution for small values of x is obtained and matched with the elastic field of a blunt crack at the elastic-plastic boundary. The crack growth criterion of critical strain is employed to determine the value of K _II of the far-field that would be required for a steadily growing crack.     

Boundary elements for three-dimensional elastic crack analysis

In this paper 8-node traction singular boundary elements are employed to represent displacement and traction variations in the vicinity of the crack front in three-dimensional geometries. The numerical procedure suggested for evaluating the singular integrals extending over these special elements is described. The efficiency and accuracy of the special elements and integration procedure are demonstrated by the results obtained in a simple test problem whose analytical solution is known. The interaction of two circular coplanar cracks embedded in an infinite medium under uniform tension loading is also analysed. Finally, the stress intensity factor variation computed for a semi-circular inner surface crack in a pressurized cylinder is presented.     

A note on quarter-point triangular elements

Recently^1?3 alternative methods of constructing triangular elements that reproduce strain singularities at crack-tips have been suggested. The differences and similarities between the approaches are considered in this note.     

A quadrature rule for conforming quadratic crack tip elements

A simple quadrature rule that precisely integrates products of singular √r and quadratic shape function gradients for a two-dimensional six node triangle and a three-dimensional fifteen node prism is derived.     

Fracture stress obtained from the elastic crack tip enclave model

The fracture stress of an elastic-plastic solid is calculated by making an approximate analysis of the crack model in which a small enclave that is surrounded by a plastic region is considered to exist at the crack tip. An attempt is made to improve Thomson's earlier analysis of this crack model by ensuring that displacements as well as traction stresses are continuous across elastic-plastic boundaries and by ensuring that the correct fracture equation is obtained for the limiting case of a perfectly elastic and a perfectly plastic solid. The fracture stress is found to increase if either or both the yield stress of the material is lowered or the rate of plastic work-hardening is reduced. It is found that the fracture stress, in contrast to Thomson's result, it is always proportional to the square root of the true surface energy of the solid.     

A warning against misuse of quarter-point elements

Quarter-point elements are used very frequently for fracture mechanics computations, because the quarter-point technique yields the required singular interpolation without any modification to existing software. This advantage is particularly significant for three-dimensional stress intensity factor computations because of the difficulty of implementing other techniques. However, in practical 3-D applications, the crack front is usually curved, and this note proves that a crack front distortion leads to a negative Jacobian in the region surrounding the crack front. The numerical difficulties to be expected depend on the aspect ratio of the elements.     

Linear elastic crack tip modelling by the displacement discontinuity method

A special crack tip element is developed to model the feature of a stress singularity at a crack tip, based on an elastic solution for an arbitrary displacement discontinuity. By incorporating this crack tip element into the conventional displacement discontinity method, the propagation of a crack with an arbitrary configuration can be modelled using the Griffith-Irwin energy criterion in a linear elastic medium. The accuracy and consistency of the method is illustrated by various examples; the results are seen to be in excellent agreement with the exact analytical solutions and the known experimental evidence. The study shows that the displacement discontinuity method is efficient and has a potential for modelling the behaviour of a complex crack.