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Journal ArticleDOI

Weight function for an elliptic crack in an infinite medium. I. Normal loading

01 Jun 2000-International Journal of Fracture (Kluwer Academic Publishers)-Vol. 103, Iss: 3, pp 227-241
TL;DR: In this paper, an integral equation method has been used to derive the crack opening displacement of an elliptic crack in an infinite elastic medium subjected to a concentrated pair of point force loading at an arbitrary location on the crack faces.
Abstract: A recently developed integral equation method has been used to derive the crack opening displacement of an elliptic crack in an infinite elastic medium subjected to a concentrated pair of point force loading at an arbitrary location on the crack faces. These results have been used to obtain the stress intensity factor along the elliptic crack front which corresponds to the weight function for an elliptic crack under normal loading. Analytical expression of the weight function can be used to derive the stress intensity factor for both polynomial loading as well as non-polynomial loading.
Citations
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Journal ArticleDOI
TL;DR: In this article, a modified procedure for the calculation of the weight function and stress intensity factor along the crack front has been proposed for different cases of the polynomial law of loading, in which the Rice energy balance equation, Dyson's theorem, and the theory of crack translation in a nonuniform stress field are used.
Abstract: For a mode-I embedded elliptical crack in an infinite elastic body, we propose a modified procedure for the calculation of the weight function and stress intensity factor based on our earlier development. Analytical and numerical values of the stress intensity factor along the crack front have been obtained for different cases of the polynomial law of loading. We propose an approach to the determination of the crack-face displacements from the stress intensity factor values, in which the Rice energy-balance equation, Dyson's theorem, and the theory of crack translation in a nonuniform stress field are used. An expression of closed form for the elliptical crack-face displacement for a polynomial law of loading of any degree has been derived, which can be employed in solving three-dimensional problems of the elasticity theory for cracked bodies.

5 citations


Cites methods from "Weight function for an elliptic cra..."

  • ...The authors of [10] proposed an original method of integral equations for solving an integral-differential equation, which relates the displacement field to the load acting on the crack surface....

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Journal ArticleDOI
TL;DR: In this article, a weight function approach is used to determine stress intensity factors for cracks in residual stress fields, due to three types of overloads for holes in metallic plates with and without subsequent remote loading.
Abstract: An understanding of how residual stresses due to overloads effect stress intensity factors for fatigue cracks at common stress concentrators, such as holes, is important. In this paper a weight function approach is used to determine stress intensity factors for cracks in residual stress fields, due to three types of overloads for holes in metallic plates with and without subsequent remote loading. The cases resulting in compressive residual stresses are remote tension overload and hole cold expansion. Also considered is the case of a tensile residual stress field due to a compressive overload. Initially generic cases for a D6ac steel plate are considered. Stress intensity solutions are given for different crack sizes for different levels of overload which produce yield zones of different sizes. For both remote overload cases it is shown that once the crack length is the same or larger than the initial yield zone, the stress intensity factors are the same as for the case without the initial overload. However, for the cold expanded hole case the beneficial reduction in stress intensity factor extends some distance outside the initial yield zone. Then the application of cold expansion to an Aluminium alloy fatigue test coupon representing a lower wing skin location in the C-130 aircraft is considered. Here the significant reduction in stress intensity factor is investigated. Some potential inaccuracies in the weight function method for compressive fields, due to possible crack closure are also discussed.

3 citations

Journal ArticleDOI
TL;DR: In this paper, the effect of Poisson's ratio on the evolution of planar sliding in the vicinity of a local drop in frictional resistance under remotely applied uniform shear load is discussed.
Abstract: Effect of Poisson’s ratio on the evolution of planar sliding in the vicinity of a local drop in frictional resistance under remotely applied uniform shear load is discussed. The local reduction of frictional resistance may be caused by various factors (for example, variability of the friction coefficient). We demonstrate that under these mixed mode conditions the shape of the slip zone may be circular for the frictional resistance with non-axisymmetric distribution and non-circular for the frictional resistance with an axisymmetric distribution. Distortion of the shape is found to be related to Poisson’s ratio. Sliding propagation in the vicinity of the parabolic drop of the frictional resistance is studied in detail.

2 citations


Cites background from "Weight function for an elliptic cra..."

  • ...In the text to follow we determine a necessary condition for the zone to remain circular during sliding and a relation between its radius a and the applied shear load T. For further convenience, a traditional polar coordinate system (p, ~o) is introduced in the plane z = 0. The stress intensity factors entering criterion of propagation (2) and calculated on the boundary of the circular zone are as follows (Borodachev, 1982, Saha and Roy, ......

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Journal ArticleDOI
TL;DR: In this article, the scattering of normally incident elastic waves by an embedded elliptic crack in an infinite isotropic elastic medium has been solved using an analytical numerical method and the representation integral expressing the scattered displacement field has been reduced to an integral equation for the unknown crack-opening displacement.
Abstract: The scattering of normally incident elastic waves by an embedded elliptic crack in an infinite isotropic elastic medium has been solved using an analytical numerical method. The representation integral expressing the scattered displacement field has been reduced to an integral equation for the unknown crack-opening displacement. This integral equation has been further reduced to an infinite system of Fredholm integral equation of the second kind and the Fourier displacement potentials are expanded in terms of Jacobi's orthogonal polynomials. Finally, proper use of orthogonality property of Jacobi's polynomials produces an infinite system of algebraic equations connecting the expansion coefficients to the prescribed dynamic loading. The matrix elements contains singular integrals which are reduced to regular integrals through contour integration. The first term of the first equation of the system yields the low-frequency asymptotic expression for scattering cross section analytically which agrees completely with previous results. In the intermediate and high-frequency scattering regime the system has been truncated properly and solved numerically. Results of quantities of physical interest, such as the dynamic stress intensity factor, crack-opening displacement scattering cross section, and back-scattered displacement amplitude have been given and compared with earlier results. ©2002 ASME

2 citations

01 Jan 2015
TL;DR: In this article, the authors investigated the coupling of the fracture modes of elliptical subsurface cracks under uniform shear loadings in two directions, and they derived the mixed mode stress intensity factors (SIFs) for cracks with aspect ratios of = 0.2-1.0 and ratios of crack depth to crack length of =0.05-1
Abstract: Original Research Paper Received 04 May 2014 Accepted 01 June 2014 Available Online 20 October 2014 Elliptical subsurface cracks are one of the probable types of cracks that occur in engineering structures, especially under rolling contact fatigue. Due to the non-symmetrical geometry, coupling of the fracture modes occurs in an elliptical subsurface crack and the crack under shear loading will experience all fracture modes. This paper investigates the coupling of the fracture modes of elliptical subsurface cracks under uniform shear loadings in two directions. First, three-dimensional parametric nite element model of crack in an in nite space has been developed and validated. Then, by moving the crack close to surface, mixed mode stress intensity factors (SIFs) have been calculated for cracks with aspect ratios of =0.2-1.0 and ratios of crack depth to crack length of =0.05-1.0. Based on the results, coupling of the fracture modes occur considerably when the crack depth becomes less than crack length. By decreasing of the crack depth from = to =0.05, shear SIFs and KI,max/KII,max ratio increase at least up to 65% and 90%, respectively. Six equations for SIFs of the subsurface cracks under uniform shear loadings in two directions have been obtained by tting to the nite element results. These equations can be used efficiently in high accurate calculation of the SIFs for subsurface cracks with any and under uniform shear loading with any direction.

2 citations

References
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Journal ArticleDOI
James R. Rice1
TL;DR: In this paper, it was shown that if the displacement field and stress intensity factor are known as functions of crack length for any symmetrical load system acting on a linear elastic body in plane strain, then the stress intensity factors for any other symmetric load system whatsoever on the same body may be directly determined.

923 citations


"Weight function for an elliptic cra..." refers background or methods in this paper

  • ...Rice (1972) independently showed that weight functions could be determined by differentiating known elastic displacement field solutions with respect to crack length....

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  • ...Foundations of the three dimensional weight function theory were given independently by Rice (1972) in the Appendix of that publication and by Bueckner (1973) in a review....

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Journal Article
TL;DR: In this paper, a state of plane strain in a notched or cracked elastic domain under the action of boundary tractions is considered, and the stress intensity factor K at a root of a notch can be re presented in the form of a weighted average of the tractions.

876 citations


"Weight function for an elliptic cra..." refers background in this paper

  • ...The concept was first introduced by Bueckner (1970) for two-dimensional cracked bodies....

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Book
01 Jan 1980

832 citations


"Weight function for an elliptic cra..." refers background or methods in this paper

  • ...Now using the following relations (see Gradshteyn and Ryzhik, 1980 )...

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  • ...Here it may be noted that the only contributing term in the weight function is W 0 0 . All other terms contribute zero either due to the orthogonal property of sine and cosine functions, or due to orthogonality of Jacobi’s polynomials or due to the relation ( Gradshteyn and Ryzhik, 1980 )...

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  • ...respect to between 0 and 1, one obtains after making use of the orthogonality property (see Gradshteyn and Ryzhik, 1980 ):...

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Book ChapterDOI
01 Jan 1983
TL;DR: In this article, the theoretical strength of crystalline solids is derived based on idealised forms of atomic force-displacement curves, in which the force is defined as the differential with respect to distance of the interatomic or inter-ionic energy.
Abstract: Calculations of the theoretical strengths of crystalline solids are usually based on idealised forms of atomic force-displacement curves, in which the force is defined as the differential with respect to distance of the inter-atomic or inter-ionic energy. The energy curve is similar to that for a diatomic molecule in that it represents the resultant of inter-atomic repulsions and attractions; the nature and strength of the attractive forces depending on the bond type: ionic, covalent, metallic, or Van der Waals. Differences in character between a lattice and a molecule occur at separations of order one lattice spacing, when Friedel oscillations in the screening charge cause the long range component of the interaction potential in the lattice to undergo a damped oscillation about zero. For small displacements, the atomic force/displacement curve is linear, having a slope equivalent to Young’s modulus, E. A lattice also has shear stiffness, denoted by the shear modulus μ. Both E and μ are defined macroscopically, usually for randomly-oriented polycrystals which are assumed to be isotropic. In single crystals, both the tensile stiffness and the shear stiffness vary with the orientation of the crystal with respect to the tensile axis and these variations can be substantial: in iron, for example, the minimum value of E is in the [100]direction (132 GPa at room temperature) and the maximum value is in the [111] direction (260 GPa).

673 citations