Stress intensity factor calculation based on the work of external forces

The external forces method is a numerical method for K calculation based on the finite element method. It uses the work of the external forces W for the calculation of the energy release rate and is particularly advantageous when that forces are applied far from the crack front. The method was applied to a corner crack geometry with the objective of studying its accuracy. Good results were obtained for a wide range of virtual crack displacements (0.03% < Δa/a < 6%) considering 4 values of W along with a polynomial regression of order 3. For that choice of parameters the inaccuracy of K is mainly due to FEM errors. A great sensitivity of K to FEM errors was observed, however accurate values of K were obtained, with errors lower than 2 percent. So, the use of the external forces method for the calculation of K is recommended, considering its simplicity and accuracy. This revised version was published online in July 2006 with corrections to the Cover Date.     

Cracks located on a single line in an orthotropic elastic medium in which body forces are acting

Plane strain problem of determining the distribution of stress in the neighborhood of cracks located on a single line is examined. The medium containing cracks is infinite, orthotropic and acted upon by an asymmetrical distribution of body forces. The crack surfaces are free from tractions and the crackline coincides with an axis of orthotropy. Fourier transform methods are employed to reduce the problem to that of solving a singular integral equation of Cauchy type.The solution is completed in the case when the infinite orthotropic medium contains a single crack. Particular distributions of concentrated loads and moments are considered in detail. The stress intensity factor and crack shape are determined and shown graphically for oak wood. The results in special and limiting cases are compared with those available in the literature.     

Stress intensity factors and COD in an orthotropic strip

The elasticity problem for an orthotropic strip or a beam with an internal or an edge crack under general loading conditions is considered. The numerical results are given for four basic loading conditions, namely, uniform tension, pure bending, three point bending, and concentrated surface shear loading. For the strip with an edge crack additional results regarding the crack opening displacements are obtained by using the plastic strip model. A critical quantity which is tabulated is the maximum compressive stress in the plane of the crack. It is shown that this stress may easily exceed the yield limit in compression and hence may severely limit the range of application of the plasticity results.

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Résumé On considère le problème d'élasticité dans une bande orthotrope ou dans une poutre présentant une fissure interne ou une fissure de bord soumise à des conditions générales de mise en charge. Des résultats numériques sont fournis dans le cas de 4 conditions de charge déterminées, à savoir la tension uniforme, la flexion pure, la flexion sur 3 points, et le cisaillement concentré en surface. Pour la bande présentant une fissure de bord, des résultats supplémentaires relatifs au COD sont obtenus en utilisant un modèle de bande plastique. On détermine une quantité critique, qui est présentée sous forme de tableau, et qui représente la contrainte de compression maximale dans le plan de la fissure. On montre que cette contrainte peut aisément dépasser la limite d'écoulement en compression et dès lors peut constituer une limitation sévère dans le domaine d'application des résultats de plasticité. Le papier inclue aussi des résultats complets pour une bande isotropique avec une fissure de bord.

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This work was supported by NASA-Langley under the Grant NGR 39-007-011 and by NSF under the Grant ENG 78-09737.

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This work was supported by NASA-Langley under the Grant NGR 39-007-011 and by NSF under the Grant ENG 78-09737.     

Stress intensity factor for a crack normal to an interface between two orthotropic materials

In this paper, the problem of a crack normal to an interface in two joined orthotropic plates is studied as a plane problem. Body force method is used to investigate dependence of the stress intensity factor on the elastic constants: E _x1, E _y1, G _xy1, V _xy1 for material 1 and E _x2, E _y2, G _xy2, V _xy2 for material 2. A particular attention is paid to simplifying kernel functions, which is used in the body force method, so that all the elastic constants involved can be represented by three new parameters: H ₁, H _2I, H ₃ for the mode I deformation and H ₁, H _2II, H ₃ for the mode II deformation. From the kernel function so obtained it is found that the effects of the eight elastic constants on the stress intensity factors can be expressed by the three material parameters, H ₁, H _2I, H ₃ and H ₁, H _2II, H ₃, respectively for K _I and K _II. Furthermore, it is also found that the dependence of K _I on H ₁, H _2I, H ₃ is exactly the same as the dependence of K _II on H ₁, H _2II, H ₃. This revised version was published online in August 2006 with corrections to the Cover Date.     

The stress intensity factor for a penny-shaped crack in an elastic body under the action of symmetric body forces

A formula is derived for the stress intensity factor at the rim of a penny-shaped crack in an infinite solid in which there is an axisymmetric distributing of body forces acting in a direction normal to the original crack surfaces. An expression for the surface displacement of the crack is also given. The use of these formulae is illustrated by a consideration of the special case in which the solid is deformed by the action of two point forces situated symmetrically with respect to the crack.

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Zusammenfassung Eine Formel für den Spannungsintensitätfaktor am Rande eines pfenninggeformten Risses in einem unendlichen Festkörper ist gewonnen. Ein achsensymmetrisches Verteilen der Körperkräite fand statt, welches in einer Richtung, normal zu der originalen Rissoberfläche wirkt. Es ist auch Ausdruck für den Oberflächenverschiebung des Risses gegeben. Die Benutzung dieser Gleichungen wird verdeutlicht durch die Betrachtung eines Spezialfalles bei dem der Festkörper durch die Wirkung zweier Punktkräfte deformiert wird, die symmetrisch zum Riss angebracht sind.

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Résumé On a établi une formule donnant le facteur de concentration de tension aux extrémités d'une fissure ferrnée disposée dans un solide infini au sein duquel une distribution de forces internes á symétrie axiale agit dans une direction normale par rapport aus surfaces de la fissure. On fournit également une expression du déplacement de ces surfaces. L'utilisation de ces formules est appliquée, à titre d' exemple, au cas spécial d'un solide soumis à l'action de deux forces concentrées symétriques par rapport à la fissure.

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This paper was prepared as a part of the work of the Applied Mathematics Research Group at North Carolina State University through the Grant AF-AFOSR-444-66 and is under the joint sponsorship of AFOSR, ARO, and ONR through the Joint Services Advisory Group.     

The stress intensity factor for a Griffith crack in an elastic body in which body forces are acting

Formulae for the stress intensity factor at the tip of a Griffith crack and for the normal component of the surface displacement are derived for a stress-free crack in an elastic solid in which there is a symmetrical distribution of body forces. Particular distributions of point forces are considered in detail.

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Zusammenfassung Eine Formel für den Spannungsintensitätfaktor an der Spitze eines Griffischen Risses und für den normal Bestandteil zu der Oberflächenverschiebung für einen spannungsfreien Riss in einem elastischen Festkörper in welchem eine symmetrische Verteilung der Körperkräfte ist, ist gewonnen. Besondere Verteilungen von Punktkräften werden ausführlich behandelt.

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Résumé Dans le cas d'une fissure qui n'est pas sous tension dans un solide élastique où existe une distribution symétrique des forces intemes, on a pu déterminer les formules donnant le facteur de concentration de tension á la pointe d'une fissure de Griffith ainsi que la composante normale du déplacement des surfaces de la fissure. On a examiné en détail certaines distributions particulières de forces concentrées.

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This paper was prepared as a part of the work of the Applied Mathematics Research Group at North Carolina State University through the Grant AF-AFOSR-444-66 and is under the joint sponsorship of AFOSR, ARO, and ONR through the Joint Services Advisory Group.     

Stress intensity factor evaluations for a curved crack in orthotropic particulate composites using an interaction integral method

This paper develops a domain-independent interaction integral (DII-integral) for extracting mixed-mode stress intensity factors (SIFs) for orthotropic materials with complex interfaces. The DII-integral does not require material property gradients, and moreover its validity is not affected by material interfaces. Combined with the extended finite element method (XFEM), the DII-integral is employed to investigate a straight crack in an orthotropic functionally graded plate and a curved crack in orthotropic particulate composites.     

The stress intensity for an edge crack in a semi-infinite orthotropic body

The problem of an edge crack in a semi-infinite plane of linear elastic orthotropic material is studied. The correction factor which relates the stress intensity factor for this problem to that for an isolated crack in an infinite body is evaluated for a range of orthotropic material properties. Calculations are restricted to mode I problems. The method requires the numerical solution of an integral equation, the integrands in which are derived from related complex variable solutions.

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Resumé On étudie le problème d'une fissure de bord dans un plan semi-infini en un matériau linéaire eastique et orthotrope. On évalue pour une gamme de propriétés orthotropes du matériau le facteur de correction qui permet de relier le facteur d'intensité de contraintes relatif à ce problème à celui relatif à une fissure isolée dans un corps infini. Ces calculs sont limités aux problèmes de rupture selon le mode. I. La méthode exige de trouver la solution numérique d'une intégrale dont les intégrants sont tirés de solutions associées à variables complexes.

     

Evaluation of stress intensity factor under pure shear stress in orthotropic material

Various types of composite materials are currently being developed and used for automobiles, airplanes, ships and other structures in response to required service conditions which are getting increasingly more severe. Of growing importance under such circumstances is the study of stress analysis and fracture mechanics for these composite material structures. Particularly, the primary concern in design of structures and machines should be the initiation of cracks due to excessive deformation, delamination in material or other material defects. In evaluating safety, it is indispensable from the structural design point of view that K value should be known by an analysis conducted in advance. In this study, stress intensity factor (mode II) under a pure shear stress was obtained using the photoelastic method and caustic method and applying an isotropic material and orthotropic material (copper fibre epoxy composite (CFEC) developed by the authors), each containing the crack. Results were compared with theoretical values. As a result, this method was found useful and the effect of the direction of the primary axis of this material on the stress intensity factor was clarified.     

Stress intensity factors for interface cracks between orthotropic and monoclinic material

Different expressions are used in the literature for the stress intensity factors of interface cracks between anisotropic material. In particular, two of these approaches will be discussed and compared for orthotropic and monoclinic materials. Relations between the stress intensity factors will be found. Expressions for the interface energy release rate G_i{mathcal{G}_i} are presented. Although the expressions appear different, they are shown to be the same by using the relations between the stress intensity factors. Phase angles are defined which may be used in a fracture criterion.     

Stress intensity factor threshold in dental porcelains

The stress intensity factor threshold (K_I0) is related to the stress level at which cracks start to grow stably, causing the weakening of porcelain prostheses during their use. The values of K_I0 of seven dental porcelains (with and without reinforcing leucite crystal, KAlSi₂O₆) stored in air (22 °C, 60% relative humidity) and artificial saliva (37 °C) were determined by measuring the crack growth velocity of radial cracks generated at the corner of Vickers indentations. The results of K_I0 were correlated with the leucite content, fracture toughness (K_Ic), and chemical composition of the porcelains. It was observed that K_I0 increased with the increase of leucite content (only for the leucite-based porcelains) and with the increase of K_Ic. The increase in Al₂O₃ content or the decrease in the alkali oxide (K₂O and Na₂O) content of the material’s glassy matrix tended to increase the K_I0 values. Storage media (air and saliva) did not significantly affect the K_I0 of porcelains tested, indicating that the control parameter of K_I0 value was not the water content of the storage media.     

Green's functions for the stress intensity factor evolution in finite cracks in orthotropic materials

The elastodynamic response of an infinite orthotropic material with finite crack under concentrated loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some example materials are obtained. This solution can be used as a Green's function to solve dynamic problems involving fini te cracks.     

Stress intensity factor solutions for finite body with quarter-elliptical flaws emanating from a notch

This paper presents a new simple engineering method for estimating the stress-intensity factor around a quarter-elliptical cracks emanating from a notch. Several examples will be used to highlight some of the advantages of the present method and several comparisons are made with other existing approaches. Stress-intensity factors produced by the present method are in good agreement with other published solutions. The present technique based on finite element analysis does not require the crack to be explicitly modelled. Making this technique ideal for studying fatigue crack growth and or shape optimisation with damage tolerance constraints.     

Generation of longitudinal eddies in a boundary layer in the presence of body forces

A combined analysis is made of measurements of the kinematic and thermal characteristics of a boundary layer disturbed by the introduction of regular threedimensional perturbations.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 57, No. 3, pp. 392–398, September, 1989.     

Stress intensity factors in finite orthotropic plates with a crack under mixed mode deformation

Mixed mode fracture problem is analyzed for the finite orthotropic plate where an inclined crack parallel to the fibre direction is centrally placed. Modified mapping collocation method with both uniform stress and uniform displacement boundary conditions is utilized to calculate stress intensity correction factors for glass/ epoxy and graphite/epoxy composites. Computed results are presented for selected combinations of crack length to width ratio L/W and plate aspect ratio H/W with various fibre orientations.     

Griffith cracks in an elastic medium in which body forces are acting

The plane strain problem of determining the distribution of stress in an infinite isotropic elastic medium containing Griffith cracks located on a single line is examined. The crack surfaces are assumed to be free from tractions, and the stress distribution in the medium is due to the action of body forces. Fourier transform methods are employed to reduce the problem to that of solving a singular integral equation of Cauchy type. The solution is completed in the case in which the medium contains a single crack. Particular distributions of concentrated loads are considered in detail, and the results are compared with those available in the literature.     

Stress intensity factor analysis of through thickness effects

The study of crack tip fields in mode I is often conducted assuming a homogeneous behaviour through the thickness. Depending on the specimen thickness, a state of plane stress or plane strain is normally presumed. However, recent studies have shown a more complex behaviour along the thickness. On the one hand, plasticity-induced crack closure effects affect mainly to a small region close to the specimen surface. On the other hand, the plastic zone evolution along the thickness is not as simple as the classic dog-bone shape normally described in Fracture Mechanics textbooks. Unlike what is normally expected, the size of the plastic zone decreases in a very small region close to the surface, as we move from the interior to the surface of the specimen. These two effects can be detected if the mesh used in the finite element model is sufficiently fine. Both effects are probably related to an uneven distribution of the load along the thickness. One of the consequences of these effects on the fatigue crack growth is the curvature of the crack front which can be explained by two mechanisms. The first one is related to the crack closure effect near the surface, it would imply a smaller effective ΔK close to the surface, and therefore a slower crack growth rate. The second one (plastic zone size decrease in a small region close to surface) is probably due to ΔK being smaller near the surface than in the interior. The current work attempts to evaluate numerically both effects in order to separate their individual influence and their magnitude. This is done by evaluating the K distribution along the thickness at different planes on an Al 2024-T35 compact tension specimen under mode I nominal loading. The plastic wake effect is removed from the model in order to distinguish between both effects.     

Crack growth in an orthotropic medium

Abstract

The elastostatic problem of an orthotropic body having a central inclined crack and subjected at infinity to a uniform biaxial load is considered. It is assumed that the crack line does not coincide with an axis of elastic symmetry of the body. The problem must be considered as one of general orthotropy, due in particular to the fact that the elastic coefficients of the material change with rotation of the reference system. The stress fields at the crack tip are reported and the presence of the non‐singular terms underlined. The Strain Energy Density Theory is extended to orthotropic materials. It is assumed that the Critical Strain Energy Density Factor has a polar variation. The crack initiation is determined via minimization of the ratio of the strain energy density over the material critical strain energy density, pointing out the effects of orthotropy and load biaxiality. The effects of the non‐singular terms on crack growth for different orthotropic materials is also studied, underling the relation between orthotropy and non‐singular terms.