Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates

Surface cracks are among the more common flaws in aircraft and pressure vessel components. Accurate stress analyses of surface-cracked components are needed for reliable prediction of their crack growth rates and fracture strengths. Several calculations of stress-intensity factors for semi-elliptical surface cracks subjected to tension have appeared in the literature. However, some of these solutions are in disagreement by 50–100%.

In this paper stress-intensity factors for shallow and deep semi-elliptical surface cracks in plates subjected to tension are presented. To verify the accuracy of the three-dimensional finite-element models employed, convergence was studied by varying the number of degrees of freedom in the models from 1500 to 6900. The 6900 degrees of freedom used here were more than twice the number used in previously reported solutions. Also, the stress-intensity variations in the boundary-layer region at the intersection of the crack with the free surface were investigated.     

Stress intensity factors and weight functions for deep semi-elliptical surface cracks in finite-thickness plates

ABSTRACT Three-dimensional finite element analyses have been conducted to calculate the stress intensity factors for deep semi-elliptical cracks in flat plates. The stress intensity factors are presented for the deepest and surface points on semi-elliptic cracks with a/t -values of 0.9 and 0.95 and aspect ratios ( a/c ) from 0.05 to 2. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were combined with corresponding results for surface cracks with a/t = 0.6 and 0.8 to derive weight functions over the range 0.05 ≤  a/c  ≤ 2.0 and 0.6 ≤  a/t  ≤ 0.95. The weight functions were then verified against finite element data for parabolic or cubic stress distributions. Excellent agreements are achieved for both the deepest and surface points. The present results complement stress intensity factors and weight functions for surface cracks in finite thickness plate developed previously.     

An approximate three-dimensional theory of layered plates containing through thickness cracks

The objective of this investigation is the development of an approximate three-dimensional theory of laminated plates for application to laminates containing through-the-thickness cracks. This is accomplished by assuming an approximate form for the stress field as a product of a function of the out-of-plane variables. The variational principle of minimum complementary potential energy is employed to obtain a system of partial differential equations and associated boundary conditions which govern the in-plane variation of the stress field. This approximate theory is then applied to the problem of a laminar composite plate containing a through-the-thickness crack subjected to in-plane loading normal to the crack face. The resulting mixed boundary value problem is solved by integral transform methods. The stress field in the vicinity of the crack edge is obtained in closed form demonstrating qualitative features characteristic of the exact three-dimensional asymptotic solution. The through-the-thickness variation of this stress field is chosen so as to enforce plane strain conditions within each layer of the composite plate.The results indicate the influence of the geometric parameters and material properties of the composite system on the amplitude and transverse variation of the stress field in the vicinity of the leading crack edge.     

Slicing procedure for approximate three-dimensional Green's functions for cracks in plates of finite thickness

Exact results for the stress intensity factor are presented for an external circular crack with oppositely directed concentrated loads applied to the crack surfaces. This result is specialized to the case of a semi-infinite crack in an infinite body with concentrated loads on the crack. A procedure is then suggested by which one can obtain from the corresponding plane result the approximate three-dimensional Green's function (concentrated load result) for any straight crack in an infinite elastic body. This procedure is used to determine the Green's functions for a finite-length crack in an infinite body, and is then used in conjunction with a suggested slicing procedure to obtain approximate three-dimensional Green's function for plates of finite thickness and infinite extent, containing finite length cracks. Previously existing solutions for crack problems are compared with results obtained by application to plate tension and bending problems of the three-dimensional Green's functions. The results indicate that the procedure yields satisfactory results when stress gradients through the plate thickness are not excessive. However, an accurate assessment of the validity of the slicing procedure awaits further progress in three-dimensional crack analysis.

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Zusammenfasung Es werden exakte Werte für den Spannungsintensitätsfaktor im Falle eines kreisförmigen Oberflächenrisses, wobei die Rißoberflächen konzentrierten und entgegengesetzten Belastungen direkt unterworfen sind. Dieses Ergebnis ist anwendbar auf den Fall eines halbunendlichen Risses in einem Körper unendlicher Abmessungen, wobei der Riß konzentrierten Beanspruchungen unterworfen ist.Anschließend wird ein Verfahren vorgeschlagen, welches es ermöglicht die dreidimensionale angenäherte Funktion von Green aus dem entsprechenden planen Ergebnis zu ermitteln und dies für den Fall eines beliebigen geraden Risses in einem elastischen Körper unendlicher Größe. Nach diesem Verfahren werden die Green'schen Funktionen für einen Riß endlicher Größe in einem unendlichen Körper bestimmt. Dieser wird anschließend dazu benutzt um mit Hilfe eines Unterteilungsverfahrens die angenäherten dreidimensionalen Green'schen Funktionen für Feinbleche unendlicher Oberfläche mit Rissen endlicher Abmessungen zu ermitteln.Die unter Anwendung der dreidimensionalen Green'schen Funktionen auf die Probleme von Zug- und Biegebe-anspruchung von Platten erzielten Ergebnisse, werden mit den schon früher vorgeschlagenen Lösungen verglichen. Dieser Vergleich zeigt, daß der vorgeschlagene Weg befriedigende Lösungen ergibt, sofern die Spannungsgradienten über die Dicke des Bleches nicht übermässig groß sind.Um jedoch die Gültigkeit des angewandten Unterteilungsverfahren exakt zu prüfen, sind bedeutende Fortschritte auf dem Gebiet der dreidimensionalen Analyse von Rissen noch erfordert.

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Résumé On présente les valeurs exactes du facteur d'intensité de contrainte dans le cas d'une fissure circulaire périphérique où des charges concentrées opposées sont directement appliquées sur ses surfaces. Ce résultat s'applique au cas de la fissure semi-infinie, dans un corps infini, des charges concentrées étant appliquées à la fissure.On suggère ensuite une procédure permettant d'obtenir la fonction tridimensionnelle approchée de Green à partir du résultat plan correspondant, et ce pour tout cas de fissure droite dans un corps infini et élastique. On détermine selon cette procédure les fonctions de Green pour une fissure de longueur finie dans un corps infini, et on l'utilise ensuite, à l'aide d'un processus de découpage, à l'obtention des fonctions tridimensionnelles approchées de Green pour des tôles d'épaisseur fine et de surface infinie, comportant des fissures de dimensions finies.Les résultats obtenus par l'application des fonctions tridimensionnelles de Green aux problèmes de traction et de flexion des plaques sont comparés aux solutions proposées antérieurement. Il résulte de cette comparaison que la procédure suivie fournit des résultats satisfaisants pour autant que les gradients de contrainte suivant l'épaisseur ne soient pas excessifs.Toutefois, pour vérifier d'une manière exacte la validité du processus de découpage qui a été adopté, il est nécessaire d'attendre que des progrès plus substantiels aient été accomplis en matière d'analyse tridimensionnelle des fissures.

     

Tension of a finite-thickness plate with a pair of semi-elliptical surface cracks

This paper deals with the tension of a finite-thickness plate with a pair of semi-elliptical cracks on both of the free surfaces. The analysis is performed in a similar manner to the previous single crack problem, by using the body force method and the boundary conditions expressed in terms of resultant forces and displacements of the boundary elements. The stress intensity factor at the maximum depth of the crack front is calculated for various values of the parameters and these results are fitted by a reliable polynomial formula for convenience of engineering applications.     

Mixed mode cracks in Reissner plates

Based on the sixth order Reissner plate theory, the generalized displacement functions for a cracked plate are derived by eigenfunction expansion method. The fractal two-level finite element method is employed to obtain the stress (moment and shear) intensity factors for the center cracked plate subjected to out-of-plane bending and twisting loads. The numerical results from the present method are checked with those available in literature. Highly accurate stress intensity factors are predicted for a wide range of thickness to crack length ratio and a full range of PoissonÆs ratio provided that the radius of fractal mesh to thickness ratio is not less than .     

Circular-arc cracks in bi-material plates under bending

On the basic of the Poisson -Kirchhoff theory of thin plates, boundary problems of bi-material plates bonded along circular arc segements are reduced to the solution of the non-homogeneous Hilbert problem in complex function theory. The unbonded portions of the interface may be regarded as flaws or crack-like imperfections of some kind. Using the properties of Plemelj formulas and Cauchy integrals, sectionally holomorphic functions are obtained for one or more cracks distributed along the circumference of a circle dividing two different materials.The results suggest the possibility of using current fracture theories to predict the failure of thin plates containing circular inserts of another material partially joined along a finite number of arcs. For illustration, stress-intensity factors employed in the Griffith-Irwin theory of fracture are computed.

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Zusammenfassung Grenzprobleme von zweimaterial Platten, die in Kriesbogenabschnitten verbunden sind, sind zu der Lösung des nicht-homogenen HILBERTproblems in der Komplexfunktiontheorie reduziert, begründet auf die POISSON-KIRCHOFF Theorie von dünnen Platten. Die unverbundenenTeileder Zwischenfläche können als Fehler oder rissähnliche Unvollkommenheiten ungewisser Art betrachtet werden. DutchGebrauch der Eigenschaften der PLEMEIJschen Formel and CAUCHY Integrale werden teilweise holomorphische Funktionen für einen oder mehrere Risse gefunden, die um den Umriss eines Kreises der zwei verschiedene Materiale teilt, verstreut sind.Die Ergebnisse deuten auf die Möglichkeit hin gegenwärtige Bruchtheorien zu gebrauchen, die das Versagen von dünnen Platten mit kreisartigea Einlagen eines anderen Materials, das teilweise entlang einer endlichen Zahl verbunden ist, voraussagen. Zur Veranschaulichung werden die in der GRIFFITHS-IRWINschen Theorie gebrauchten Spannungsintensitätsfaktoren berechnet.

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Résumé Sur la base de la théorie de POISSON — KIRCHHOFF sur les plaques minces, les problèmes aux limites de plaques constituées de deux matériaux différents assemblés jointivement le long de segments d'arc circulaires sont ramenés à la solution du problème non homogène de HILBERT exprimé suivant la théorie des fonctions complexes.Les portions de l'interface des tôles où existe une solution de continuité peuvent être considérées comme des défauts ou des imperfections assimilables à des fissures.En utilisant les propriétés des formules de PLEMELJ et des intégrales de CAUCHY, on obtient les fonctions holomorphes relatives à une ou plusieurs fissures distribuées suivant la circonférence d'un cercle séparant deux matériaux différents.Les résultats suggèrent la possibilité de recourir aux théories courantes de la rupture pour prédire la rupture de plaques rpinces comportant des inclusions circulaires d'un autre matériau que le materiau de base, mais qui loi seraient partiellement solidarisées le long d'un nombre d'arcs fini. A titre d'illustration, on a calculé les facteurs de concentration de contraintes que l'on utilise dans la théorie de la rupture de Griffith — Irwin.

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Presented at the Fifth United States National Congress of Applied Mechanics, University of Minnesota, Minneapolis, Minnesota, June 14–17, 1966.     

Inelastic response of axisymmetric plates with cracks

The behavior of stepped annular plates made of a fiber reinforced composite material is studied making use of the method of mode form motions. The plates having stable cracks of prescribed deepness at the re-entrant corners of steps are subjected to an initial impact and the subsequent motion is due to inertia. The designs of stepped plates are established which minimize maximal residual deflections under given material consumption. Numerical results are presented for the plate with single step.     

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     

Stress intensity factors for cracks in thin plates

This paper presents stress intensity factor solutions for several crack configurations in plates. The loadings considered include internal pressure, and also combined bending and tension. The dual boundary element method is used to model the plate and mixed mode stress intensity factors are evaluated by a crack surface displacement extrapolation technique and the J-integral technique. Several cases including centre crack, edge crack and cracks emanating from a hole in finite width plates are presented.     

Equations of growth of fatigue cracks in inhomogeneous plates

We propose a computational model for the determination of the kinetics of propagation of fatigue cracks in plates whose material has inhomogeneous mechanical characteristics and fatigue fracture resistance. We also take into account the history of plastic deformation of the plate. The model is based on the energy criterion of fatigue fracture of materials. We deduce a system of differential equations describing the kinetics of fatigue cracks in inhomogeneous plates and present the solutions of two particular problems obtained by using the proposed model. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 3, pp. 53–58, May–June, 1999.     

Stress intensity factors for three-dimensional cracks

The method of superposition of analytical and finite-element solutions is proposed for determining three-dimensional distributions of the stress intensity factor; the singular part of the solution is expressed by a linear combination of analytical solutions, and the rest by a finite-element solution. The method is applied to a round bar with a circumferential crack and plates with penetrating cracks. Detailed distributions of the stress intensity factor near the plate surfaces are investigated with the aid of Benthem's theory, which shows that less than 0.5% of the plate thickness is severely influenced by the plate surfaces in the case of a compact tension specimen. Computations for the present method can be performed with a general purpose program for finite element analysis without using special elements.

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Résumé La méthode de superposition des solutions analytiques et par éléments finis est proposée pour déterminer les distributions tri-dimensionnelles du facteur d'intensité des contraintes; la partie singulière de la solution est exprimée par une combinaison linéaire des solutions analytiques et le reste de la solution par une solution à éléments finis. La méthode est appliquée aux cas d'une barre ronde comportant une fissure circonférentielle, et de tôles comportant des fissures pénétrantes. Les distributions détaillées du facteur d'intensité de contrainte au voisinage des surfaces de la tôle sont analysées à l'aide de la théorie de Benthem, qui montre que moins de 0,5% de l'épaisseur de la tôle est sévèrement influencée par l'effet de surfaces dans le cas d'éprouvette de tension compacte. Les calculs de la méthode présentée peuvent être exécutés à l'aide d'un programme à objectifs généraux destiné à l'analyse par élément fini sans recourir à des éléments spéciaux.

     

Fracture of composite plates with periodical edge cracks

The elastostatic properties of two sets of periodically arranged and perfectly bonded strips are investigated under the assumption that every set of strips has edge cracks perpendicular to the interfaces, and that the external loads are applied parallel to the interfaces far away from the crack region. The problem is formulated in terms of a system of singular integral equations and solved numerically to obtain the stress intensity factors.

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Résumé On étudie les propriétés élasto-statiques de deux séries de feuillards arrangés de manière périodique et parfaitement collés, sous l'hypothèse que chaque série de feuillards présente des fissures de bord perpendiculaires à l'interface et que les charges externes sont appliquées parallèlement à ces interfaces et suffisamment loin de la zône de fissuration. Le problème est formulé en terme d'un système d'équations intégrales singulières et résolu numériquement en vue d'obtenir les facteurs d'intensité de contrainte.

     

Analyses of infinite pairs of surface cracks in elastic plates

Stress intensity factors and crack opening displacements are presented for infinite pairs of surface cracks in plates subjected to remote tension by using the three dimensional weight function method developed in [7,8]. A wide range of configuration parameters is considered. The results compare very well with double edge cracks as crack aspect ratio tends to zero; with collinear cracks as it tends to infinity; with a pair of surface cracks in a wide plate when the ratio of crack length to plate width is small; and with a single surface crack in large plates when both the ratios of crack length to plate width and crack depth to plate thickness are small. Also illustrated is the significant difference between a single surface crack and the surface cracks in pairs when the ratio of crack depth to plate thickness is large.     

Extension of generalized plane strain plates with reinforced cracks

Applying the transversally isotropic generalized plane strain plate of Mindlin and Kane, as refined by Kotousov and Wang, the authors use Fourier transforms to develop a hyper-singular line-spring model for a reinforced crack under extensional loads, enabling the analysis of cracks in plates of arbitrary thickness. The new model is used to develop fracture mechanics geometry correction factors to account for the effect of the transverse modulus, Poisson’s ratio, and plate thickness on the stress intensity and crack opening displacement. The model is then used to examine the reinforcing effect of springs bridging the crack faces and extend the Rose model to include plate thickness effects, allowing the inclusion of plate thickness effects in the fracture mechanics analysis of reinforced cracks.     

A three-dimensional analysis of laminated orthotropic plates

A mathematical apparatus is developed for the analysis of the three-dimensional stress–strain state with small deflections of laminated orthotropic plates of any thickness and non-symmetrical layer structure through the thickness. This theory is based on the representation of the displacement vector in terms of products of the given functions in the direction of the axes x₁, x₂ and the unknown functions over the axis x₃. Both the real and complex roots of the characteristic equation describing the representation of the sought functions through the thickness of the plate are taken into account. Such an approach allows us to expand the scope of physical and mechanical relationships among the material characteristics.