Simplified Greens functions for mode I and II cracks

When using integral equation techniques to solve fracture mechanics problems, a solution for the stress and displacement fields surrounding an arbitrarily loaded crack face within an infinite two-dimensional plane can be used to accurately represent the behavior of the crack. It is advantageous to have this solution in as simple a form as possible, since the accuracy and computation time for numerical solution of the integral equations is improved by using less complex kernels. Previous solutions for the Green's functions for loaded cracks in an infinite domain involve lengthy complex variable expressions or double integrals with Cauchy and crack tip singularities. The results presented here are a simplification of a previous double integral solution, and the simplified result is a single integral representation with a non-singular integrand which involves only real numbers. Both the mode I and mode II problems are reduced to this simplified form, and the two may be combined for mixed mode two-dimensional crack problems.

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Résumé Lorsqu'on utilise la technique de l'équation intégrale pour solutionner des problèmes de mécanique de rupture, on peut utiliser une solution pour les champs de contraintes et de déplacement environnant la surface d'une fissure arbitrairement mise en charge dans un plan infini à deux dimensions, pour représenter, de manière sûre, le comportement de la fissure. Il est avantageux que cette solution soit la plus simple possible en ce qui regarde sa forme, car la précision et le temps de calcul des solutions numériques des équations intégrales sont améliorés en utilisant des kernels moins complexes. Les solutions précédemment proposées pour les fonctions de Green relatives à des fissures chargées dans un domaine infini comportent de longues expressions de variables complexes ou de doubles intégrales à singularité de Cauchy, ainsi que des singularités d'extrémités de fissure. Les résultats présentés ici constituent une simplification de la solution précédente à double intégrale, et le résultat simplifié en est une représentation à simple intégrale avec un intégrant non singulier qui comporte seulement des nombres réels. Les problèmes de Mode I et de Mode II sont réduits à cette forme simple et ces deux problèmes peuvent être combinés pour des problèmes de fissures à deux dimensions, sollicitées selon un mode mixte.

     

Computation of shear mode weight functions for circular and elliptical cracks

Three-dimensional shear mode fundamental fields in finite bodies with mixed boundary conditions are analyzed by a special finite element method for circular and elliptical cracks. A procedure for determining the Fourier coefficients of the stress intensity factor for circular cracks is presented. A special series is proposed to represent the computed crack face weight functions for elliptical cracks.     

Point weight function method application for semi-elliptical mode I cracks

The method of the approximate weight function construction for a semi-elliptical crack was suggested. The weight function sought was written as the sum of asymptotic (weight function for an elliptical crack in an infinite body) and correction components. To take into account the influence of a body free surface on the asymptotic component behavior, fictitious forces symmetric with respect to the body free surface were introduced.As an example of the efficiency of the proposed method semi-elliptical axial cracks in pressure vessels were considered. The results of the stress intensity factor prediction are in good agreement with the corresponding results obtained by Raju and Newman. The only exception are the results for the points located near the major ellipse axis. This may be explained by the shortcomings of the employed empirical weight function expression for an elliptical crack in an infinite body.     

Mode I stress intensity factors for curved cracks in gears by a weight functions method

The most recent trend in power transmission design considers the damage-tolerant approach as one of the methods to obtain safe, reliable and light systems. This means that components containing cracks must be considered and analysed to understand the conditions that cause critical cracks and defects and their dimensions.
In this paper a cracked tooth of an automotive gearwheel is considered. A numerical procedure (based on the slice synthesis weight function method) to calculate the stress intensity factors of curved cracks due to bending loads is illustrated. The results are compared with those obtained by expensive finite element calculations. The agreement is satisfactory and the proposed technique proves to be very attractive from the point of view of time saving.
One example of an application to fatigue design practice is provided, namely the analysis of fatigue crack propagation in surface-treated gears. The results show the role played by residual stresses induced by carburizing and shot peening.     

A closed-form method for integrating weight functions for part-through cracks subject to Mode I loading

Numerical integration of weight functions tends to be computationally inefficient because of the singularity in a typical weight function expression. An alternative technique has been developed for surface and corner cracks, which greatly improves both efficiency and accuracy of K_I estimates. Exact analytical solutions for the weight function integral are obtained over discrete intervals, and then summed to obtain the stress intensity factor. The only numerical approximation in this approach is the way in which the variation in stress between discrete known values is treated. Closed-form weight function integration methods are presented for three approximations of the stress distribution: (1) constant stress over each integration interval, (2) a piecewise linear representation, and (3) a piecewise quadratic fit. A series of benchmark analyses were performed to validate the approach and to infer convergence rates. The quadratic method is the most computationally efficient, and converges with a small number of integration increments. The piecewise linear method gives good results with a modest number of stress data points on the crack plane. The constant-stress approximation is the least accurate of the three methods, but gives acceptable results if there are sufficient stress data points.     

Determination of approximative weight functions for straight through cracks

The method of determining approximative weight functions, well known for straight through surface cracks (edge cracks), is extended to internal straight through cracks. The procedure is outlined in detail for central cracks in endless strips of infinite and finite widths. For the finite width strip the coefficients of the weight function are given in a closed form.

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Résumé On étend aux fissure internes traversantes et directes la méthode de détermination des fonctions pondérales approximatives, dont l'application aux fissures droites débouchant en surface (fissures de bord) est bien connue.On décrit en détail la procédure utiisée dans le cas de fissures centrales dans des bandes sans fin de largeur infinie be finie.Pour les bandes de largeur finie, les coefficients de la fonction pondérale sont fournis sous une forme fermée.

     

Calculation of weight functions for three-dimensional bodies with cracks

A new algorithm is proposed for the numerical calculation of weight functions used in the determination of stress intensity factors by the finite elements method. The algorithm is based on the method of equivalent volume integration. It is shown that weight functions can be obtained for cracks in three-dimensional bodies with the stress intensity factor averaged over a small section of the crack front. A numerical example demonstrating the usefulness of the algorithm is presented.Moscow. Moscow Physico-Engineering Institute. Translated from Problemy Prochnosti, No. 11, pp. 32–36, November, 1989.     

Computation of three-dimensional weight functions for circular and elliptical cracks

Three-dimensional mode I fundamental fields for circular and elliptical cracks in isotropic, finite bodies with prescribed displacement and traction boundaries are analyzed by a previously introduced finite element method [1]. For the circular crack, we present a procedure for determining the Fourier coefficients of the stress intensity factor by using the ordinary fundamental fields.

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Résumé En recourant à une méthode par éléments finis présentée précédemment (1), on analyse les champs fondamentaux à trois dimensions de Mode I correspondant à des fissures circulaires et elliptiques dans des corps finis isotropes, qui sont soumis à des limites définies de déplacements et de traction.Dans le cas d'une fissure circulaire, on présente une procédure pour déterminer les coefficients de Fourier du facteur d'intensité de contrainte, en utilisant les champs fondamentaux ordinaires.

     

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

The weight functions of the configuration of collinear cracks

Collinear cracks have been the subject of many investigations and publications. This paper provides (once and for all) the weight functions for the general configuration. The functions are equally useful for modes I and II.     

A mode II weight function for subsurface cracks in a two-dimensional half-space

The general properties of a mode II Weight Function for a subsurface crack in a two‐dimensional half‐space are discussed. A general form for the WF is proposed, and its analytical expression is deduced from the asymptotic properties of the displacements field near the crack tips and from some reference cases obtained by finite elements models. Although the WF has general validity, the main interest is on its application to the study of rolling contact fatigue: its properties are explored for a crack depth range within which the most common failure phenomena in rolling contact are experimentally observed, and for a crack length range within the field of short cracks. The accuracy is estimated by comparison with several results obtained by FEM models, and its validity in the crack depth range explored is shown.     

Computation of weight functions for three-dimensional cracks by the boundary element method

A boundary element procedure is presented for calculating weight functions for three-dimensional cracks with a smooth front. The weight function for the particular point at the crack front is represented as a sum of regular and singular parts. The known weight function for a circular crack in an infinite body is used as the singular part. The boundary integral equation is formulated for the regular part of the weight function in the vicinity of considered crack front point and for the whole weight function for the rest of the body. A discretized form of the boundary integral equation is given. Some examples are provided to test the accuracy of the proposed procedure.     

First and second order stress extrapolation for mode I and mode II edge cracks

The use of linear and second order stress extrapolation to obtain K_I and K_II in two-dimensional finite element models of a thick plate containing an edge crack was examined. Three loading cases were studied, including classical Mode I and Mode II problems and a problem of tribological contact. Linear extrapolation was observed to yield accurate predictions of K_I in cases of dominant Mode I loading. In Mode II situations, notably where the crack faces experienced compressive normal stresses, second order extrapolation was observed to improve estimates of K_II.     

Stress intensity factors for tunnelling corner cracks under mode I loading

Stress intensity factors (SIFs) presented in the literature for corner cracks are limited to ideal quarter-circular and quarter-elliptical crack shapes. This paper presents SIF solutions for corner cracks that exhibit tunnelling, extending the range of corner crack shapes illustrated in the literature. Solutions were developed in a parametric form, obtained by empirically fitting polynomials to numerical values of SIF obtained from the FEM. A parameter was defined to quantify the extent of tunnelling. It was observed that crack shape has a significant effect on the SIF, so the consideration of equivalent quarter-circular cracks can produce inaccurate results when significant tunnelling occurs. SIF solutions for quarter-circular cracks are also presented and compared with those quoted in the literature.     

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.     

The crack opening displacement field of semi-elliptical surface cracks in tension for weight functions applications

Application of the weight function method for calculating energy release rates or averaged weighted stress intensity factors requires that both the stress intensity factors and the crack opening displacements are known for a reference load case. This report gives an approximative solution for the crack opening displacement field of a semi-elliptic surface crack under pure tension loading. As a practical example the energy release rates are calculated for bending and compared with results available in the literature. Also a test procedure is described for checking the quality of approximative stress intensity factors.

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Résumé Pour appliquer la méthode des fonctions pondérées au calcul des vitesses de relaxation de l'énergie ou des facteurs d'intensité de contraintes moyens pondérés, il faut que soient connus les facteurs d'intensité de contraintes et les déplacements d'ouverture de la fissure, dans un cas de sollicitation de référence. La rapport fournit une solution approximative pour le champ de COD correspondant à une fissure superficielle semi-elliptique soumise à une charge de traction pure. Comme exemple pratique, on calcule les vitesses de relaxation de l'énergie correspondant à la flexion, et on se compare avec les résultats publiés dans la littérature. On décrit également une procédure d'essai en vue de vérifier la qualité des facteurs d'intensité de contraintes approchés.

     

The standard model in cavity quantum electrodynamics. I. General features of mode functions for a Fabry-Perot cavity

Abstract

In the present and the accompanying paper a justification of the standard model of cavity quantum electrodynamics is given in terms of a quasi-mode theory of macroscopic canonical quantization. The coupling of the cavity quasi-mode to external quasi-modes is treated for the representative case of the three-dimensional Fabry-Perot cavity. The general form of the travelling and trapped mode functions for this cavity are derived in this paper and the mode-mode coupling constants are calculated in the accompanying paper. The slow dependence of the coupling constants with the mode frequency difference demonstrates that the conditions for Markovian damping of the cavity quasimode are satisfied. As also discussed in the accompanying paper, the interaction of radiative atoms with cavity quasi-modes is associated with reversible energy exchanges between atom and cavity and represented by Rabi coupling constants. The interaction of radiative atoms located within the cavity with sideways travelling external quasi-modes involves slowly varying coupling constants and is associated with irreversible spontaneous emission dampling. The basic processes represented in the standard cavity quantum electrodynamics model and the associated coupling constant and decay rates thereby follow from the quasi-mode theory.     

Analysis of temperature effects near mode I cracks in glassy polymers

A previous isothermal study (Estevez et al., Journal of Mechanics and Physics of Solids 48, 2585–2617, 2000) has shown that the toughness of glassy polymers is governed by the competition between shear yielding and crazing. The present work aims at investigating loading rates for which thermal effects need to be accounted for. The influence of the heat coming from the viscoplastic shear yielding and from crazing on their competition and on the toughness is examined. Crazing is shown to be the dominant heat source, and the dependence of the craze properties on temperature appears to be key in controlling the toughness of the material.*Author for correspondence (E-mail: rafael.estevez@insa-lyon.fr)     

Some applications of the weight functions theory to the analysis of cracks

The Bueckner-Rice weight functions theory and the weight functions computational methods are studied.For the multiparametric configurations a system of equations is obtained which allows the stress intensity factors to be calculated for arbitrary loads on the basis of the reference solutions without solving the boundary-value problem.A possibility of the linear spring model application to the weigh functions calculation is discussed.     

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.