Elastic-plastic analysis of surface flaws using a simplified line-spring model

The line-spring model has proven to be an effective tool for evaluating fracture parameters in surface-cracked plates and shells. However, application of the model requires detailed numerical computations, necessitating the availability of a specialized computer code. For approximate engineering calculations a version of the model which is more convenient to implement computationally, would be useful.In this paper a simplified line-spring model is presented along with detailed illustration of its application. The simplification is accomplished by replacing the crack front with a crack of constant depth and treating the ligament “spring” as elastic perfectly plastic. Despite its simplicity the model gives reasonably accurate predictions of fracture parameters, such as the J-integral or crack opening displacement (COD) at the root of surface cracks. This will be demonstrated by comparing analytical results for J and COD with previously published experimental data for surface-cracked steel plates.     

Analysis of surface cracks using the line-spring boundary element method and the virtual crack extension technique

The authors have developed a new line-spring boundary element method which couples the line-spring model with the boundary element method to deal with the problem of a surface cracked plate. However, the drawback of the line-spring model is that a reliable stress intensity factor could not be directly obtained near the free surface intersection. Therefore, the virtual crack extension technique is employed in a post-processor of the line-spring boundary element method to obtain the stress intensity factor at the crack front-free surface intersection. Theoretical analysis is described. Stress intensity factors for surface cracks are calculated to verify the proposed method. The interaction of two surface cracks is also investigated. The solutions obtained by the line-spring boundary element method show that the method proposed is efficient and reasonably accurate.     

Weight functions for edge cracks in thin surface layers

A procedure is given for the calculation of the weight function for a surface crack located in a thin surface layer perpendicular to the interface. The procedure is demonstrated for cracks approaching the interface and cracks going beyond the interface.     

Weight function method for interface cracks in anisotropic bimaterials

This paper extends the Bueckner-Rice weight function method to interface crack problems in anisotropic bimaterials. This method allows one to use a known crack solution to determine any other solutions for the same geometry. It is shown that, other than a different moduli matrix, the interface weight function relation is formally identical to that in the homogeneous case, and many existing results for homogeneous crack analysis can be directly applied to the interface crack problem. For collinear cracks between two elastic half-spaces satisfying a non-oscillatory condition, a correspondence principle is established so that an anisotropic crack solution could be obtained by simply vectorizing the corresponding isotropic solution.     

Experimental verification of the simplified line-spring model

The recently developed simplified line-spring model is employed to obtain estimates of the crack opening displacement (COD) in surface cracked plates in tension and circumferentially cracked pipes in pure bending. The predictions of the model for the plate are compared to experimental data obtained from a series of tests on surface cracked X70 line pipe steel plates, on the other hand the results from the model for the pipe are compared to previously presented experimental and theoretical results. The simplified model seems to offer estimates of the COD which are in reasonably good agreement with both experimental data and existing theoretical results.

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Résumé On utilise le modèle simplifié dit line-spring — qui consiste à simuler les forces de rétention de propagation d'une fissure de surface dans le ligament restant, en dispensant des efforts et des moments sur une fissure traversante de même longueur — en vue d'estimer le déplacement d'ouverture d'une fissure dans des tôles fissurées en surface et soumises à traction, et dans des tubes fissurés circonférentiellement et soumis à flexion pure.Les prédictions du modèle dans le cas d'une tôle sont comparées aux données expérimentales recueillies dans une série d'essais sur des tôles fissurées en surface d'acier pour pipe-line X70. Par ailleurs, les prédictions du modèle dans le cas du tube sont comparées à des résultats théoriques et expérimentaux obtenus précédemment.Le modèle simplifié utilisé parait offrir des estimations du COD en assez bon accord avec les résultats expérimentaux d'une part, et l'application des théories existantes d'autre part.

     

Dynamic stress intensity factors analysis of interface crack using line-spring model

In this study, a new method for calculating the dynamic stress intensity factors of a bimaterial bending specimen with an interface crack is proposed by making use of a line-spring model. A pre-cracked bending specimen is modeled by one-dimensional beam finite elements and a line-spring representing the stiffness or compliance of a cracked part. The proposed method enables the one-dimensional analysis of a two-dimensional crack problem; thus the time variations of the dynamic stress intensity factors of a bimaterial bending specimen with an interface crack can be obtained by making use of a personal computer within a few minutes. The results obtained by the proposed method agree reasonably well with those obtained by the two-dimensional finite element method, although a slight difference in period can be found. The proposed method enables rapid evaluation of dynamic stress intensity factors. So a rapid evaluation system of the dynamic fracture toughness of a bimaterial with an interface crack can be achieved by combining an instrumented impact test apparatus with a computer program based on the proposed method which runs on a personal computer.     

Weight functions for circular cracks

The authors examined a problem for a linearly elastic solid containing internal or external circular normal separation crack. It is shown that the corresponding weight function is equal to the product of the axisymmetric weight function and Poisson's kernel. An approximate weight function for an internal circular crack in an unlimited cylinder is constructed as an example.Translated from Problemy Prochnosti, No. 12, pp. 25–28, December, 1990.     

Weight functions for interface cracks

Weight functions are developed for determining stress intensity factors of cracks along an interface between two linear, elastic materials. As a result of the interface, both mode I and II components will be present for all but very special loading cases. The weight functions are employed to produce exactly the known stress intensity factors of a crack along an interface loaded by tensile and shear point forces.Part of this work was carried out while the author was on sabbatical leave at the Materials Laboratory, Wright Patterson Air Force Base, Ohio, USA     

Weight function based Dugdale model for mixed-mode crack problems with arbitrary crack surface tractions

Weight function theory states crack surface displacements can be found for any arbitrary distribution of mode I, or mixed-mode crack face tractions via that geometry’s weight functions. This statement is validated via finite element analysis of the infinite center-cracked plate for various mixed mode loadings. An elastic-perfectly plastic material is considered using a Dugdale approach and compared to elastic-plastic finite element simulations. The weight function method in all cases agrees well with the finite element simulations for small scale yielding at the crack tip. As the maximum traction value approaches one-half the yield strength discrepancies become larger due to violation of small scale yielding.     

Line-spring model for surface cracks in a reissner plate

In this paper the line-spring model developed by Rice and Levy for a surface crack in elastic plates is reconsidered. The problem is formulated by using Reissner's plate bending theory. For the plane strain problem of a strip containing an edge crack and subjected to tension and bending new expressions for stress intensity factors are used which are valid up to a depth-to-thickness ratio of 0.8. The stress intensity factors for a semi-elliptic and a rectangular crack are calculated. Considering the simplicity of the technique and the severity of the underlying assumptions, the results compare rather well with the existing finite element solutions.     

A closed-form solution to the equivalent through-crack model for surface cracks

Summary A closed-form solution for plates containing surface cracks is obtained by using an equivalent through-crack model, which reduces the three-dimensional problem of the surface cracked plates to a two-dimensional Hilbert problem. The effect of surface crack is replaced by a continuous distribution of forcesN(x, 0) and momentsM(x, 0) applied along the crack face of the equivalent through-crack model. A convenient form of expressing these forces and moments is by using power polynomials. Then the singular integral equation, expressing the solution of the Hilbert problem can be readily integrated.According to this model we assumed that the crack depth at extremities, where the crack intersects the free surface of the plate, is not zero. This assumption, corroborated by experimental evidence, means that a singularity exists at the extremities of the crack. The experimental evidence was achieved by using the method of caustics and photoelasticity. The results of the evaluation of the stress intensity factor for elliptical cracks were compared well with the solutions of the respective 3-D problem solved by applying the finite-element method, the line-spring model based on the Reissner plate theory, the finite-element alternating method and the benchmark estimate. The distribution of the stress intensity factor along the crack lips, as calculated in this paper, was dropping off rapidly in the surface layer and it was very close to the results given by the approximate 3-D theory.With 7 Figures     

Two-parameter fracture mechanics and circumferential crack growth in surface cracked pipelines using line-spring elements

A procedure for constraint correction of crack growth resistance curves for single edge notched specimens and for pipe geometries is presented. The procedure is based on FE models with the combination of shell- and line-spring finite elements. Crack tip opening displacement and T-stress are employed, and ductile crack growth is accounted for. Experimental crack growth resistance curves are obtained for both single edge notched tension- and bending-specimens for different crack depths to cover significantly different constraint levels. To account for different constraint levels, a method to scale the resistance curve using the T-stress is implemented. The analyses include ductile crack growth in both the circumferential and thickness directions. The effect of circumferential crack growth with biaxial loading is also presented. The results from the line-spring model are compared with detailed 3D-models for verification of the implementation of circumferential crack growth. The importance of including crack growth in circumferential direction is discussed based on numerical parametric studies. A measure to quantify the importance of circumferential crack growth is proposed.     

Weight functions for bimaterial interface cracks

An efficient approach using the analytically decoupled near-tip displacement solution for bimaterial interface cracks presented in this paper involves: (1) the calculation of the decoupled strain energy release rates G _I and G _II associated respectively with the decoupled stress intensity factors K _I and K _II and (2) the extension of Rice's displacement derivative representation of Bueckner's weight function vectors beyond the homogeneous media. It is shown that the stress intensity factors for a bimaterial interface crack predicted by the present approach agree very well with those solutions available in the literature. The computational efficiency is enhanced through the use of singular elements in the crack-tip neighborhood.As reported in the homogeneous case, the calculated weight function for a bimaterial interface crack is load-independent but depends strongly on geometry and constraint conditions. Due to the coupling nature of the stress intensity factors of a bimaterial interface crack, the invariant characteristics of the dimensionless weight function vectors are different from those of a crack in homogeneous material. In addition, the elastic constants of two constituents can significantly alter the weight function behavior for a cracked bimaterial medium.Due to the load-independent characteristic of the weight functions, the stress intensity factors for a bimaterial interface crack can be obtained accurately and inexpensively by performing the sum of worklike products between the applied loads and the weight functions for the cracked bimaterial body under any loading conditions once the weight functions are explicitly predetermined. The same calculation can also be applied for the identical cracked bimaterial medium with different constraint conditions by including the self-equilibrium forces that contain both the external loads and the reaction forces induced at the constraint locations. Moreover, the physical interpretation of the weight functions can provide a guidance for damage tolerant design application.     

Determination of crack surface displacements for cracks emanating from a circular hole using weight function method

Cracks emanating from a circular hole are of significant engineering importance, especially in aerospace industry. Accurate determination of key fracture mechanics parameters is essential for damage tolerance design and fatigue life predictions. The purpose of this paper is to provide an efficient and accurate closed‐form weight function approach to the calculation of crack surface displacements for radial crack(s) emanating from a circular hole in an infinite and finite‐width plate. Results were presented for two loading conditions: remote applied stress and uniform stress segment applied to crack surfaces, and extensively compared to recent studies using other methods in the literature. Both single and double radial cracks were considered, and also the effect of finite plate width on crack surface displacements has been investigated. A brief assessment was made on an engineering estimation of displacements based on a correction of stress intensity factor ratio. It has been demonstrated that the Wu‐Carlsson closed‐form weight functions are very efficient, accurate and easy‐to‐use for calculating crack surface displacements for arbitrary load conditions. The method will facilitate fatigue crack closure and other fracture mechanics analyses where accurate crack surface displacements are required.     

A weight function methodology for the assessment of embedded and surface irregular plane cracks

A low cost numerical tool for the calculation of mode I stress intensity factors K in embedded and surface irregular cracks is presented in this paper. The proposed tool is an extension of the O-integral algorithm due to Oore and Burns for the assessment of embedded plane cracks using the weight function methodology. The performance of the O-integral is assessed first by comparing its K results to exact solutions for embedded elliptical and rectangular cracks. From the analysis of this data it is found that the error in the K results systematically depends on the crack aspect ratio and the local crack front curvature. Based on this evidence a corrective function is derived in order to remediate the limitations of the O-integral. Solutions due to Newman and Raju are used to account for the effects of free surfaces and finite thickness. The accuracy of the proposed procedure is assessed by solving a number of examples and by comparing the obtained results to those available in the literature.     

Weight functions for cracks in finite rectangular plates

A numerical method for the determination of weight functions relevant to cracked bodies with finite dimensions is presented. The discretized nodal weight function is determined by means of the finite element method, and the assumed form of the crack-face weight function is successfully demonstrated by using the least squares fitting procedure. The explicit weight functions of mode I are presented for plates of finite width and length with single edge, double edge and center cracks. With the use of the uncracked stress field in the crack faces and the crack-face weight function, the efficient calculation of the stress intensity factor is illustrated. The size of the Dugdale plastic zone ahead of the crack tip for a finite plate is estimated from the available weight functions. Some practical examples for various crack configurations and loading systems are given and a very satisfactory degree of accuracy is obtained from the results when compared with the findings of earlier studies.

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Résumé On présente une méthode numérique pour la détermination des fonctions pondérales relatives à des corps fissurés de dimensions finies. Au moyen de la méthode des éléments finis, on détermine la fonction correspondant à une discrétisation nodale, et la forme supposée de la fonction relative à la surface de la fissure se trouve confirmée grâce à une procédure de positionnement par moindres carrés. On présente les fonctions pondérales explicites de Mode 1 pour des tôles de largeur et longueur finies, comportant des fissures sur un seul bord, sur deux bords ou au centre. En appliquant aux faces de la fissure le champ de constraintes en condition non fissurée et la fonction pondérale relative à ces faces, on montre comment calculer efficacement le facteur d'intensité de contrainte. Les fonctions disponibles permettent d'estimer la taille de la zone plastique de Dugdale en avant de la fissure dans le cas d'une tôle de grandeur finie. On fournit quelques exemples pratiques correspondant à diverses configurations de fissures et de modes de sollicitation. Quand on compare les résultats obtenus à ceux résultant d'autres approches, on est frappé par le degré de précision auquel la méthode permet d'accéder.

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Notation a crack length - E Young's modulus - v Poisson's ratio the prospective crack plane - K ⁽¹⁾ stress intensity factor for load system (1) - K ⁽²⁾ stress intensity factor for load system (2) - T ⁽²⁾ (s, a) surface traction on the boundary - s arc length of the boundary - u(¹⁾ (s, a) displacement field on the tractionS ₁ - h (s, a) weight function on the traction boundaryS ₁ - x distance from the crack mouth - T ⁽²⁾ _y (x, a) stresses in the crack faces normal to the prospective crack plane - u ⁽¹⁾ _y (x, a) crack opening displacement - h (x, a) weight function on the crack faces - remote stress - _y yield stress - c Dugdale plastic zone size - W width of plate - L length of plate boundary