Dynamic stress intensity factor due to concentrated loads on a propagating semi-infinite crack in orthotropic materials

The elastodynamic response of an infinite orthotropic material with a semi-infinite crack propagating at constant speed under the action of concentrated loads on the crack faces is examined. Solution for the stress intensity factor history around the crack tip is found for the loading modes I and II. Laplace and Fourier transforms along with the Wiener-Hopf technique are employed to solve the equations of motion. The asymptotic expression for the stress near the crack tip is analyzed which lead to a closed-form solution of the dynamic stress intensity factor. It is found that the stress intensity factor for the propagating crack is proportional to the stress intensity factor for a stationary crack by a factor similar to the universal function k(v) from the isotropic case. Results are presented for orthotropic materials as well as for the isotropic case.     

The stress intensity factor of an edge crack in a finite elastic disc

In this paper, a Mellin transform technique is used to express the stress intensity factor and the crack energy of an edge crack in a finite elastic disc directly in terms of the solution of a Fredholm integral equation of the second kind. The constant loading case is considered in detail and the results given in graphical form.     

Thermal stress intensity factors of an infinite orthotropic layer with a crack

Stress intensity factors are determined for a crack in an infinite orthotropic layer. The crack is situated parallel to the plane surfaces of the layer. Stresses are solved for two kinds of the boundary conditions with respect to temperature field. In the first problem, the upper surface of the layer is heated to maintain a constant temperature T ₀, while the lower surface is cooled to maintain a constant temperature –T ₀. In the other problem, uniform heat flows perpendicular to the crack. The surfaces of the crack are assumed to be insulated. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the crack, the difference in temperature at the crack surfaces and differences in displacements are expanded in a series of functions that vanish outside the crack. The unknown coefficients in each series are evaluated using the Schmidt method. Stress intensity factors are then calculated numerically for a steel layer that behaves as an isotropic material and for a tyrannohex layer that behaves as an orthotropic material.     

The stress intensity factor of an edge crack in a finite rotating elastic disc

In this paper the problem of determining the stress intensity factor and crack formation energy of an edge crack in a finite rotating elastic disc is reduced to the solution of a Fredholm equation. Numerical results are given.     

Stresses in an orthotropic semi-infinite strip due to edge loads

Summary The stresses in an orthotropic elastic semi-infinite strip subject to plane strain are investigated. Symmetrical distributions of surface tractions are prescribed on the sides of the strip, while along the end the boundary conditions are arbitrary. By using an integral transform method the problem is reduced to a singular integral equation. The dependence of the stress singularity and the stress-intensity factors on the orthotropic properties of the strip is investigated. Stress distributions over the strip end are evaluated numerically.With 5 Figures     

Determination of thermal stress intensity factors for an interface crack in a graded orthotropic coating-substrate structure

The thermal fracture problem of an interface crack between a graded orthotropic coating and the homogeneous substrate is investigated by two different approaches. For the case that most of the material properties in the graded orthotropic coating are assumed to vary as an exponential function, the integral transform and singular integral equation technique is used to obtain some analytical results. In order to analyze the case with more complex material distribution, an interaction integral is presented to evaluate the thermal stress intensity factors of cracked functionally graded materials (FGMs), and then the element-free Galerkin method (EFGM) is developed to obtain the final numerical results. The good agreement is obtained between the numerical results and the analytical ones. In addition, the influence of material gradient parameters and material distribution on the thermal fracture behavior is also presented.     

Anti-plane shear problem for an edge crack in a finite orthotropic plate

The problem of an edge crack in a finite orthotropic plate under anti-plane shear is considered. The boundary collocation method is used to calculate the mode III stress intensity factor (SIF). For the case in which the material is isotropic, the present results agree very well with those obtained by using the integral equation method. Furthermore, the method can be extended readily for general cases with arbitrary geometrical and boundary loading conditions and material properties.     

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.     

Wedge loading of a semi-infinite strip with an edge crack

The problem of a semi-infinite strip containing an edge crack is considered. It is assumed that the strip is loaded by a frictionless rigid wedge pressed into the crack. The resulting crack-contact problem is formulated in terms of a system of singular integral equations. The behavior of the solution near the singular points is studied in detail. A series of numerical examples is given and the results are compared with those obtained by the method of boundary collocation and by the simple beam theory.

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Résumé Le problème d'une tôle mince semi-infinie contenant une fissure latérale est considérée. On suppose que le feuillard est soumis à une charge par un coin rigide et sans friction appliqué dans la fissure. On formule le problème du contact de fissure qui en résulte en termes d'un système d'équations intégrales singulières. Le comportement de la solution correspondant aux points singuliers est étudié dans le détail. Une série d'exemples numériques est fournies et les résultats sont comparés avec ceux obtenus par la méthode de collocation des frontières et par la théorie simple des poutres.

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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 77-19127.     

The practical evaluation of stress intensity factors at semi-infinite crack tips

The solution of crack problems in plane or antiplane elasticity can be reduced to the solution of a singular integral equation along the cracks. In this paper the Radau-Chebyshev method of numerical integration and solution of singular integral equations is modified, through a variable transformation, so as to become applicable to the numerical solution of singular integral equations along semi-infinite intervals, as happens in the case of semi-infinite cracks, and the direct determination of stress intensity factors at the crack tips. This technique presents considerable advantages over the analogous technique based on the Gauss-Hermite numerical integration rule. Finally, the method is applied to the problems of (i) a periodic array of parallel semi-infinite straight cracks in plane elasticity, (ii) a similar array of curvilinear cracks, (iii) a straight semi-infinite crack normal to a bimaterial interface in antiplane elasticity and (iv) a similar crack in plane elasticity; in all four applications appropriate geometry and loading conditions have been assumed. The convergence of the numerical results obtained for the stress intensity factors is seen to be very good.     

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.     

Magnetic stress intensity factor for an edge crack in a soft ferromagnetic elastic half-plane under tension

Summary This paper applies the theory for magnetoelasticity to solve the plane problem of an edge crack in a soft ferromagnetic half-plane subjected to a far-field tension and a uniform magnetic field. Fourier transform techniques are used to formulate the mixed boundary value problem as a singular integral equation. The stress intensity factor is calculated and is shown graphically. Tensile tests are also performed on a cracked ferromagnetic plate with strain gage technique, and the numerical results are compared with the test data.     

Stress intensity factors for an inclined edge crack in a semiplane

Mode I and II Stress Intensity Factors under uniform general biaxial loadings were derived for an inclined edge crack in a semiplane. By interpolating Finite Element results in the angular range [0°÷80°], analytical expressions were obtained for both K_I and K_II with an accuracy better than 1%. Influence coefficients were defined in the crack reference frame thus highlighting the coupling effects between Modes I and II due to the loss of symmetry when the crack is not normal to the surface.     

The effective crack length of an edge slot in a semi-infinite sheet under tension

The rate of change of the elastic strain energy with respect to notch depth is considered for edge notches or slots with a finite constant radius, , at the tip. Numerical results are obtained for an edge slot in a semi-infinite sheet under tension. It is found that the slot can be considered as a crack of length L + 1.18 for this purpose. In finite sheet, this result is valid for small L/ ratios.

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Zusammenfassung Das Wechselverhältnis der elastischen Dehnungsenergie zur Kerbentiefe wird fuer Randkerben oder Schlitze mit endlichem konstanten Radius, , an der Spitze betrachtet. Numerische Ergebnisse werden für einen Randschlitz in einer halb-unendlichen Platte unter Zugspannung gefunden. Es wird gefunden, dass zu diesem Zwecke der Schlitz als Riss mit Länge L + 1.18 betrachtet werden kann. Für eine endliche Platte ist dieses Resultat für kleine L/ Verhältnisse gültig.

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Résumé L'evolution de l'énergie de deformation elastique en fonction de la profondear d'entaille est etudiée dans le cas d'entailles lateriales ou de saignées .....dont le rayen de courbure d'extrémité a une valeur finie et constante.Des données numeriques oat été oblénues dans le cas d'une saignée effectivée sur les bord d'un feuillard semi infini soumis à tension. On trouve que la saignee peut-être dans ce cas assimilee a une fissure qui aurait une longueur L + 1.18 . En ce qui concerne le cas de la tole finie, ce resultat n'est applicable que pout de faibles valeurs du rapport L/.

     

Elastodynamic stress intensity factors for a semi-infinite crack due to three-dimensional concentrated shear loadings on the crack faces

The dynamic stress intensity factors for a semi-infinite crack in an otherwise unbounded elastic body is investigated. The crack is subjected to a pair of suddenly-applied shear point loads on its faces at a distance l away from the crack tip. This problem is treated as the superposition of two problems. The first problem considers the disturbance by a concentrated shear force acting on the surface of an elastic half space, while the second problem discusses a half space with its surface subjected to the negative of the tangential surface displacements induced by the first problem in the front of the crack edge. A fundamental problem is proposed and solved by means of integral transforms together with the application of the Wiener–Hopf technique and Cagniard–de Hoop method. Exact expressions are then derived for the mode II and III dynamic stress intensity factors by taking integration over the fundamental solution. Some features of the solutions are discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of an edge crack in a semi-infinite composite with a long reinforced phase

For the purpose of clarifying the micro fracture of continuous fiber unidirectionally reinforced composite materials, the problem of an edge crack perpendicular to a long reinforced phase is considered on the basis of the plane strain theory of elasticity. The stress intensity factor at the tip of the crack, and the stresses on the interface between the matrix and the reinforced phase and in the reinforced phase are discussed. In the analysis, the method of continuous distributions of dislocations is used. Then, a singular integral equation is derived and is solved by the technique developed by Erdogan and Gupta. From the numerical results it was concluded that:

1. The stress intensity factor decreases monotonically as the crack tip approaches the reinforced phase. That is, the presence of the reinforced phase can result in crack arrest.
2. When the crack tip exists near the reinforced phase, the normal stress on the interface between the matrix and the reinforced phase has a maximum at the intersection of the extension of the edge crack and the reinforced phase, while the shear stress on the interface and the normal stress in the reinforced phase take, respectively, maxima at symmetric points with respect to the crack surface in the immediate vicinity of the intersection.
3. The maximum values of the stresses on the interface and in the reinforced phase increase monotonically as the crack tip approaches the reinforced phase.

     

Stress intensity factors for an elliptical crack approaching the surface of a semi-infinite solid

Stress intensity factors for an embedded elliptical crack approaching the free surface of the semi-infinite solid that is subjected to uniform tension perpendicular to the plane of crack are presented in a nondimensional form for various crack aspect ratios and crack distances from the free surface. Stress intensity factors are determined numerically using an alternating technique with two solutions. The first solution involves an elliptical crack in a solid and subjected to normal loading expressible in a polynomial of x and y. The second solution involves stresses in the half space due to prescribed normal and shear stresses on the surface. Effect of the Poisson's ratio on these stress intensity factors is also investigated. Stress intensity factors for a semi-elliptical surface crack in a tinite thickness plate are then estimated in a nondimensional form for various crack aspect ratios and crack depth to plate thickness ratios.Specialist Engineer, Aerospace Group, The Boeing Company, Seattle, Washington.Professor, Department of Mechanical Engineering, University of Washington, Seattle, Washington, and also Aerospace Group, The Boeing Company, Seattle.     

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.