Evaluation of the stress field around a notch tip using contour integrals

The mode I and mode II asymptotic stresses around a notch tip are in general governed by different orders of singularity. Direct computation of the mixed-mode near-tip stress field therefore appears to be difficult. In this paper, we propose a pair of contour integrals J_kR. The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated with finite element solutions. As an aside, by defining a pair of generalized stress intensity factors (SIFs) (K_I)_β and (K_II)_β, the relationship between J_kR and the SIFs is derived and expressed as functions of the notch angle β. Once the J_kR-integrals are accurately computed, the generalized SIFs and, consequently, the asymptotic mixed-mode stress field can then be properly determined. The feasibility of our formulation is demonstrated in two numerical examples, where various instances with different notch angles are considered. No particular singular elements are used in this study.     

A general treatment of creep crack growth

A theoretical model for creep crack based on energy balance criterion in the fracture process region is proposed in the present paper. A concept based on the dissipation of thermoelastic-plastic-creep flux into the fracture process region is introduced from which a generalized power integral C_g^* was derived. This integral when is used in conjunction with the tearing modulus and other material parameters can characterize the crack growth behaviour in solids subject to general thermomechanical loadings. The analytical results computed by the proposed model have shown excellent agreement with some experimental results published by other prominent researchers.     

A near optimal crack tip mesh

The near optimal crack tip mesh obtained by the authors in a previous paper is tested for a set of problems. The accuracy is maintained within the desired values. The direct displacement extrapolation method for SIF estimation is compared to the virtual crack extension. The methodology for crack tip mesh design discussed previously is validated.     

A mixed mode crack tip finite element

A special finite element for plane analysis of elastic structures with through the thickness cracks is presented. Generalized displacements are used in developing the element, and it contains the proper singularities. The opening (K _I), in plane shearing (K _II), and combined (K _I+K _II) modes of deformation are present. The stiffness matrix is given explicitly and its eigenvalues are shown. Numerical results are presented and compared with other solutions.

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Résumé On présente un système spécial d'éléments finis pour l'analyse en état plan de structures élastiques comportant des fissures traversant l'épaisseur du produit. Les déplacements généralisés sont utilisés pour le développement de ces éléments, ceux-ci comportant leurs propres singularités. On présente l'ouverture (K _I) en cisaillement plan (K _II) et en modes combinés de déformation (K _I+K _II). La matrice de rigidité est donnée explicitement et on montre qu'elle correspond à eigenvalues. Les résultats numériques sont présentés et comparés avec d'autres solutions existantes.

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This work was sponsored by Martin-Marietta's Independent Research and Development Program.     

A methodology for crack tip mesh design

A methodology for crack tip mesh design is developed which consists of comparing the mesh geometric parameters against the accuracy of the finite element solution. By successive changes in the mesh parameters a near optimal mesh can be obtained. This was done here two-dimensional linear elastic single mode problems. The direct displacement extrapolation method for stress intensity factor estimation is used.     

A special bending crack tip finite element

The general expression of the Williams expansion around a bending crack tip is presented. Based on this a special circular bending crack tip finite element of an arbitrary high order is developed, and the element stiffness matrix is explicitly given, which is especially convenient for engineering analyses. Numerical examples are presented and compared with previous results to demonstrate the efficiency and accuracy of the special element.     

Calculation of mixed-mode stress intensity factors for a crack normal to a bimaterial interface using contour integrals

A pair of contour integrals J_kε are proposed in this paper. The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated without using any particular singular finite elements. Also, the relationship between J_kε and the generalized stress intensity factors (SIFs) is analytically derived and expressed as functions of the bimaterial mechanical constants. Once the J_kε-integrals are accurately computed, the generalized SIFs and, consequently, the asymptotic mixed-mode stress field can then be properly determined. Numerical results in this study show that the contribution from mode II stress component appears to be more dominant when the uncracked material is relatively stiffer, and vice versa.     

A general procedure transferring domain integrals onto boundary integrals in BEM

This paper is concerned with transferring to the boundary the domain type integrals occurring in the boundary integral approach applied to boundary value problems with nonzero body force terms. The framework used encloses many interesting engineering applications, e.g. elastostatics, heat conduction and magnetostatics. Beside this are pseudo plastic strains are incorporated due to interaction phenomena with relevant quantities. The proposed method is based upon the homogenization of the governing differential equation by using a particular solution of the inhomogeneous one. Various methods deriving such particular solutions are considered.     

Semi-empirical crack tip analysis

Experimentally observed crack opening displacements are employed as the solution of the multiple crack interaction problem. Then the near and far fields are reconstructed analytically by means of the double layer potential technique. Evaluation of the effective stress intensity factor fesulting from the interaction of the main crack and its surrounding crazes in addition to the remotely applied load is presented as an illustrative example. It is shown that crazing (as well as microcracking) may constitute an alternative mechanism to Dugdale-Barenblatt models responsible for the cancellation of the singularity at the crack tip.

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Résumé Pour solutionner le problème d'interactions multiples des fissures, on utilise les COD observées expérimentalement. On reconstitue ensuite analytiquement les champs proche et lointain à l'aide de la technique du potentiel en deux couches. Comme exemple illustratif, on présente une évaluation du facteur effectif d'intensité de contraintes résultant de l'interaction d'une fissure principale avec les brisures qui l'entourent, ainsi que de la charge appliquée à distance. On montre que le phénomène de brisures-comme celui de microfissuration-constitue une alternative aux modèles de Dugdale-Barenblatt auxquels on doit la suppression de la singularité à l'extrémité de la fissure.

     

A generalization of Elliott's model of a crack tip

Elliott's model of a crack tip is generalized to encompass a wider range of force laws. The associated crack extension criterion is shown to be equivalent to that obtained using the classic Griffith equation, and the relation between the generalized Elliott model and the cohesive zone model is also discussed. The original Elliott model, which is based on the stress and displacement distribution in the vicinity of a sharp crack tip, appears as a special case.     

Effect of crack closure on non-linear crack tip parameters

Crack closure concept has been widely used to explain different issues of fatigue crack propagation. However, some authors have questioned the relevance of crack closure and have proposed alternative concepts. The main objective here is to check the effectiveness of crack closure concept by linking the contact of crack flanks with non-linear crack tip parameters. Accordingly, 3D-FE numerical models with and without contact were developed for a wide range of loading scenarios and the crack tip parameters usually linked to fatigue crack growth, namely range of cyclic plastic strain, crack tip opening displacement, size of reversed plastic zone and total plastic dissipation per cycle were investigated. It was demonstrated that: (i) LEFM concepts are applicable to the problem under study; (ii) the crack closure phenomenon has a great influence on crack tip parameters decreasing their values; (iii) the ΔK_eff concept is able to explain the variations of crack tip parameters produced by the contact of crack flanks; and (iv) the analysis of remote compliance is the best numerical parameter to quantify the crack opening level. Therefore the crack closure concept seems to be valid. Additionally, the curves of crack tip parameters against stress intensity factor range obtained without contact may be seen as master curves.     

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.     

A tensile crack in creeping solids with large damage near the crack tip

An asymptotic analysis of stationary mode I crack in creeping solids with large damage near crack tip is conducted. To consider the damage effect, Kachanov damage evolution law is utilized and incorporated into the power-law creep constitutive equation. With the compatibility equation, a nonlinear eigenvalue problem which can be solved by numerical approaches is established. From this result, the distribution of stress and strain rate are obtained with the coupling effect of damage and creep under plane stress condition. Also the influence of material parameters on the stress is examined. According to the result, it is shown that the creep exponent n and damage parameter (=/(1+k)) have a significant effect on determining the eigenvalue s and angular distribution of stress and strain rate near the crack tip. The creep exponent n plays the role to soften and damage parameter plays the role to harden the material near the crack tip. The stress and strain rate show quite different behavior from those of HRR problem.     

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.     

The structure in the vicinity of a crack tip: A general theory based on the cohesive zone model

Consideration is given to the development of a general theory for the determination of the structure in the vicinity of the tip of a crack within an isotropic elastic solid. The basis of the model is the presence of a Barenblatt-type zone, in which cohesive stresses act, thereby allowing non-linear behaviour to he taken into account. The theory enables the stress and displacements within the cohesive zone, the zone size and critical crack extension stress to be determined, for a wide range of stress-displacement laws describing the non-linear behaviour within the cohesive zone, provided its size is small compared with that of the crack.     

Time dependent crack tip enrichment for dynamic crack propagation

We study several enrichment strategies for dynamic crack propagation in the context of the extended finite element method and the effect of different directional criteria on the crack path. A new enrichment method with a time dependent enrichment function is proposed. In contrast to previous approaches, it entails only one crack tip enrichment function. Results for stress intensity factors and crack paths for different enrichments and direction criteria are given.     

Monotonic and cyclic crack tip plasticity

Monotonic and cyclic plastic zone sizes were measured in a medium strength ferrite-pearlite steel (BM 45) tested in fatigue at 25 Hz at room temperature. Two methods were applied: microhardness and the recently developed ‘fatigue in compression’ technique. The results obtained are discussed in terms of accuracy and reliability.The retardation effect due to overloads was also studied in the same material and is illustrated experimentally as a $\text{d}\text{a}\text{d}\text{N}$ vs ΔK curve. This effect emphasizes the importance of an accurate evaluation of both the size and shape of the overall plastic zone. The shape and dimensions of the cyclic plastic zones seem to indicate that in ductile metals the steady state of fatigue crack growth occurs under plane strain conditions.