Crack-growth resistance of laminated hybrid SMC composites

The present work is aimed at characterizing the crack-growth behavior of hybrid composites having two different stacking sequences, namely; [C% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaaigdadaWcba% WcbaGaaGymaaqaaiaaikdaaaaaaa!39AA!\[1\tfrac{1}{2}\]R/C/% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaaigdadaWcba% WcbaGaaGymaaqaaiaaikdaaaaaaa!39AA!\[1\tfrac{1}{2}\]R/C] and [R/C/% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaaleaaleaaca% aIXaaabaGaaGOmaaaaaaa!38EF!\[\tfrac{1}{2}\]R/C/% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaaleaaleaaca% aIXaaabaGaaGOmaaaaaaa!38EF!\[\tfrac{1}{2}\]R/C/R]. The letter C in these hybrids represents a 50% weight fraction continuous fiber composite layer while the R represents a 50% weight fraction randomly oriented chopped fiber composite layer. The standard compact tension specimens having notches oriented along and perpendicular to the fiber direction of the C layers were chosen. The flexural strength is purported to be an adequate parameter for crack-growth characterization in the cases where the observed crack-growth was in the direction perpendicular to the notch direction. For the cases of self-similar crack-growth, the measured fracture resistance was found to be lower than that predicted by a proposed rule-of-mixture relationship based upon the crack-energy release rates of the C layers and R layers.

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Résumé Le but de ce travail est de caractériser le comportement à la croissance de la fissure de composites hydrides constitués de deux séquences d'empilement différentes, à savoir C% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaaigdadaWcba% WcbaGaaGymaaqaaiaaikdaaaaaaa!39AA!\[1\tfrac{1}{2}\]R/C/% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaaigdadaWcba% WcbaGaaGymaaqaaiaaikdaaaaaaa!39AA!\[1\tfrac{1}{2}\]R/C et R/C/% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaaleaaleaaca% aIXaaabaGaaGOmaaaaaaa!38EF!\[\tfrac{1}{2}\]R/C/% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaaleaaleaaca% aIXaaabaGaaGOmaaaaaaa!38EF!\[\tfrac{1}{2}\]R/C/R. La lettre C de ce composite hybride représente une couche comportant 50% en poids de fibres composites continues et R représente une couche de fibres composites comportant 50% en poids de fibres fractionnées et réparties au hasard. On a choisi des éprouvettes de traction compactes présentant des entailles orientées soit parallèlement soit perpendiculairement à la direction des fibres des couches C. Dans les cas où l'on a observé une croissance de fissure dans une direction perpendiculaire à la direction de l'entaille, on a constaté que la résistance à la flexion était un paramètre adéquat pour caractériser cette croissance de fissure. Dans certains cas de croissance de la fissure, on a constaté que la résistance à la rupture était inférieure aux prédictions lorsque celles-ci étaient établies sur une proposition de règle de mélange, basée sur les taux de relaxation d'énergie de fissuration dans les couches C et dans les couches R.

     

Mode I equivalent state of crack-like defects in multiaxial stress fields

The novel concept of equivalent state randomly oriented flaws developed from the generalized fracture toughness theory [1] is presented. Based on this concept, planar defects located in multiaxial stress field regions, characterized by modes I, II, and III stress intensity factor combinations, are distinguished by a mode I equivalent state stress intensity factor % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadUeagaqeaa% aa!3846!\[\bar K\]₁ of identical function. Accordingly, the complex mode fracture criterion is exactly replaced by the conventional mode I criterion % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadUeagaqeaa% aa!3846!\[\bar K\]₁ K _1C . It is demonstrated that this criterion is mathematically equivalent to other more complex generalized fracture criteria [2,4,5], i.e., it predicts the same critical conditions.Current approximate procedures applied to crack-like defects detected in structural components, based on reorienting or orthogonally projecting the defect over a plane normal to the maximum principal tensile stress, are discussed and applied to two simple structural applications. When the results are compared with those from the proposed equivalent state flaw method, it is concluded that, to a large extent, the procedures are inconsistent and generate significant errors that may lead to incorrect decisions over the remaining service life of the structure.The equivalent state flaw concept is used to establish the equivalent state mode I threshold value % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadUeagaqeaa% aa!3846!\[\bar K\]₁ corresponding to complex stress state fatigue loadings.

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Résumé On présente le concept original de défauts équivalents répartis au hasard, développé à partir de la théorie généralisée sur la ténacité à la rupture [Réf. 1]. Sur base de ce concept, des défauts plans situés dans des zones à champs de contrainte multi-axiale, et caractérisés par des facteurs d'intensité de contrainte combinant les modes I, II et III, sont caractérisés par un facteur d'intensité de contrainte équivalent % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadUeagaqeaa% aa!3846!\[\bar K\]₁; relatif à un mode I et occupant la même fonction. Dès lors, le critère décrivant la rupture sous un mode complexe est en tous points remplacé par le critère conventionnel % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadUeagaqeaa% aa!3846!\[\bar K\]₁ K _1C .Ce critère est mathématiquement equivalent aux autres critères généralisés de rupture, de forme plus complexe [Réf. 2, 4, 5], en ce qu'il prédit les mêmes conditions critiques.On discute, et on applique à deux cas de structures simples, les procédures habituelles d'approximation pour des défauts assimilables à des fissures détectés dans des composants. Ces défauts sont réorientés ou projetés orthogonalement sur un plan normal à la plus grande tension principale.Lorsqu'on compare les résultats de ces procédures d'approximation à ceux que fournit la méthode proposée, on en conclut que ces procédures sont, dans une large mesure, incorrectes, et qu'elles donnent lieu à des erreurs importantes susceptibles de conduire à des décisions erronées sur la vie résiduelle d'une structure.Le concept de défaut équivalent est utilisé pour établir une valeur critique équivalente % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadUeagaqeaa% aa!3846!\[\bar K\]₁ en mode I, correspondant à le seuil des sollicitations de fatigue de mode complexe.

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Operated for the U.S. Department of Energy, Contract No. DE-AC12-76-N0052.     

On strength scaling of composites with long aligned fibres

Strength measurements of a specially made composite material are reported. The material consisted of an epoxy matrix and the reinforcement of one layer of well aligned and equally spaced glass fibers. For a certain range of fiber spacing , matrix initiation strength _c, under uniaxial tension, and were related as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZTGaam% 4yaOWaaOaaaeaacqaH7oaBaeqaaiabgIKi7UGaae4saaaa!3E6C!\[\sigma c\sqrt \lambda \approx {\text{K}}\]. Dimensional analysis indicated that the constant was proportional to the matrix fracture toughness. Using reported data on matrix initiation strength of a borosilicate glass-SiC composite under bending, a similar relation was suggested between _c and the maximum fiber spacing, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZTGaam% 4yaOWaaOaaaeaacqaH7oaBjeaycaqGTbGaaeyyaiaabIhaaOqabaGa% eyisISlaaa!40D6!\[\sigma c\sqrt {\lambda {\text{max}}} \approx\]. The latter relationship was also applicable to hardened cement paste with the largest void on the fracture surface as the appropriate length scale. The results outlined in this paper suggest that, for a range of fiber spacing, matrix initiation strength is dictated by a Griffith type flaw proportional to the fiber spacing (uniform case) or the largest fiber spacing (non-uniform case).     

Three-dimensional finite element calibration of the short-rod specimen

The short rod is a simple, inexpensive configuration for fracture toughness testing. Since no rigorous analytical stress-intensity factor calibration of this geometry has yet appeared, a three-dimensional finite element study was undertaken. Successively finer meshes were employed to investigate convergence in compliance versus crack length, and the dimensionless calibration constant, A, in the expression% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Teada% WgaaWcbaacbaGaa4xmaiaa-ngaaeqaaOGaeyypa0Jaa8NramaaBaaa% leaacaWFJbaabeaakiaa-feacaGFVaGaa43waiaa-jeadaahaaqcba% uabeaalmaalyaajeaqbaGaaG4maaqaaiaaikdaaaaaaOGaaiikaiaa% igdacqGHsislcaWF2bWaaWbaaSqabKqaafaacaWFYaaaaOGaaiykam% aaCaaaleqajeaqbaWaaSGbaeaacaaIXaaabaGaaGOmaaaaaaGccaGG% Dbaaaa!4A74!\[K_{1c} = F_c A/[B^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} (1 - v^2 )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ]\]Quarter-point singular elements were used along the crack front, and a range of crack lengths 0.65 <- a/B <- 1.1 was investigated. Distributed and point loading cases were considered. Polynomials were least-squares fit through the compliance data and were differentiated to yield expressions for average stress-intensity factor along the crack front% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Teada% WgaaWcbaacbaGaa4xmaaqabaGccqGH9aqpdaWadaqaamaalaaabaGa% a8NramaaCaaaleqajeaybaGaa8Nmaaaakiaa-veacaWFNaaabaGaaG% Omaiaa-jgacaWFGaaaamaalaaabaGaa4hzaiaa-neaaeaacaqGKbGa% a8xyaaaaaiaawUfacaGLDbaadaahaaWcbeqcbauaamaalyaabaGaaG% ymaaqaaiaaikdaaaaaaaaa!477B!\[K_1 = \left[ {\frac{{F^2 E'}}{{2b }}\frac{{dC}}{{{\text{d}}a}}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \]Minima of this expression were obtained corresponding to critical average stress-intensity factors and crack lengths. The above expressions were then equated to solve for calibration constant values of,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-feacq% GH9aqpcaaIYaGaaGynaiaac6cacaaI5aGaaeiiaiabgglaXkaabcca% caqGXaGaaeOlaiaabkdacaqG1aaaaa!4227!\[A = 25.9{\text{ }} \pm {\text{ 1}}{\text{.25}}\]at a _c/B=0.86 for the distributed load case and% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-feacq% GH9aqpcaaIYaGaaGimaiaac6cacaaI5aGaaeiiaiabgglaXkaabcca% caqGXaaaaa!4004!\[A = 20.9{\text{ }} \pm {\text{ 1}}\]at a _c/B=.69 for the point load case.The distributed load case value of A is in very good agreement with previously reported values of 24.4 ± 1.3 and 25.0, and is about 11 percent higher than the currently recommended value obtained through K _Ic correlation.Preliminary results of a study of stress-intensity factor variation along the crack front are also presented. They show that maximum stress-intensity factor occurs at the edges of the crack front. This observation is consistent with the reverse tunneling phenomenon sometimes observed in short rod testing.Recommendations for further numerical study of the short rod configuration are suggested.

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Résumé Le barreau court est d'une configuration simple et peu coûteuse pour les essais de ténacité à la rupture. Comme il n'est pas encore apparu de calibrage rigoureux du facteur d'intensité de contrainte relatif à cette géométrie, on a entrepris une étude par éléments finis à trois dimensions. Un maillage de plus en plus fin a été utilisé pour étudier la convergence entre la compliance et la longueur de fissuration ainsi que la constante de calibrage sans dimension A, dans l'expression de K _Ic.Le long du front de fissuration, on a eu recours à des éléments singuliers en quart point et on a examiné une gamme de longueurs de fissure rapportées au diamètre et comprise entre 0,65 et 1,1. On a envisagé des cas de mises en charge réparties et ponctuelles.Les valeurs polynomiales ont été ajustées par la méthode des moindres carrés grâce aux donnés de compliance et ont été différenciées en vue d'aboutir à des expressions d'un facteur d'intensité de contrainte moyen le long du front de fissure.Les minima de la valeur K _I ainsi trouvés ont été obtenus en faisant se correspondre les facteurs d intensité critiques moyens de contrainte et les longueurs de fissuration, on a pu ainsi définir les valeurs de constante de calibration en faisant se correspondre K _Ic et K _I, et l'on a tiré% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-feacq% GH9aqpcaaIYaGaaGynaiaac6cacaaI5aGaaeiiaiabgglaXkaabcca% caqGXaGaaeOlaiaabkdacaqG1aaaaa!4227!\[A = 25.9{\text{ }} \pm {\text{ 1}}{\text{.25}}\]pour a _c/B=0,86 correspondant à une mise en charge distribuée et% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-feacq% GH9aqpcaaIYaGaaGimaiaac6cacaaI5aGaaeiiaiabgglaXkaabcca% caqGXaaaaa!4004!\[A = 20.9{\text{ }} \pm {\text{ 1}}\]pour a _c/B=0,69 dans le cas d'une mise en charge ponctuelle.La valeur de A correspondant à la mise en charge répartie est en excellent accord avec les valeurs précédemment trouvée 24,4 ± 1,3 et 25,0 et est d'environ 11% supérieure à la valeur généralement recommandée obtenue par une corrélation avec K _Ic.On présente également les résultats préliminaires d'une étude de la variation des facteurs d'intensité de contrainte le long du front de fissuration. Ces résultats montrent que le facteur d'intensité des contraintes passe par un maximum aux bords du front de la fissure. Cette observation est conforme à un phénomène inverse à celui du phénomène tunnel, qui est rencontré parfois dans les essais de barreau court.On suggère des recommandations pour une étude numérique complémentaire de la configuration du barreau court.

     

On singular integral equations for kinked cracks

Chebyshev polynomial techniques for solution of singular integral equations leading to square root singularities at the ends of the interval of integration are studied. It is shown that the results are less accurate when a singularity, albeit a weak one, appears between the interval ends. Typical examples are problems involving kinked cracks. Some attempts to improve the accuracy are discussed.

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Résumé On considère l'application de la technique polynominale de Chebyshev pour solutionner les intégrales singulières conduisant à des singularités d'ordre % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaamaalyaabaGaaG% ymaaqaaiaaikdaaaaaaa!38EB!\[{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\] aux extrémités de l'intervale d'intégration. On montre que les résultats sont moins précis lorsqu'une singularité, même faible, apparait entre les extrémités de l'intervale. Les problèmes comportant des fissures tortueuses constituent des exemples types d'application. On discute de diverses tentatives pour améliorer la précision des résultats.

     

The effects of blowing and suction on free convection boundary layers on vertical surfaces with prescribed heat flux

The effects that blowing and suction have on the free convection boundary layer on a vertical surface with a given surface heat flux are considered. Similarity equations are derived first, their solution being dependent on the wall flux exponentn and a dimensionless transpiration parameter , (as well as on the Prandtl number). The range of existence of solutions is considered, with it being shown that solutions exist only forn > –1 for blowing,whereas they exist for alln >n ₀ for suction, wheren ₀ < –1 and depends on . The solutions for strong suction and blowing are derived. In the latter case the asymptotic structure is found to be different forn in the three ranges –1 <n < – % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3775!\[\frac{1}{4}\], –% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3775!\[\frac{1}{4}\] <n < % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabgkHiTmaaleaaleaacaaIXaaabaGaaGinaaaaaaa!3EB1!\[ - \tfrac{1}{4}\], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabgkHiTmaaleaaleaacaaIXaaabaGaaGinaaaaaaa!3EB1!\[ - \tfrac{1}{4}\] <n < % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaI3aaabaGaaGOmaaaaaaa!3DC8!\[\tfrac{7}{2}\],n % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaaleaaleaacaaI3aaabaGaaGOmaaaaaaa!3DC8!\[\tfrac{7}{2}\]. Results are then obtained for the non-similarity problem of constant heat flux with a constant transpiration velocity. Solutions valid for large distances from the leading edge for both suction and blowing are derived.     

Experimental and numerical studies in model composites Part I: Experimental results

Results on strength, apparent toughness, fatigue crack growth and fiber debonding on specially made composite materials are reported. The compact tension composite specimen used consisted of an epoxy matrix and layers of long aligned glass or kevlar fibers that were equally spaced. The experimental data on crack initiation strength showed that for a range of fiber spacing , the composite's strength _A , scaled with the fiber spacing in the form of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa% aaleaaieGacaWFbbaabeaakmaakaaabaGaeq4UdWgaleqaaOGaeyyp% a0JaeqOUdSgaaa!3EB5!\[\sigma _A \sqrt \lambda = \kappa \]. The apparent toughness of the composite specimens increased with a decrease in fiber spacing. Two sets of fatigue crack propagation experiments were performed. The first one was on specimens with the same fiber spacing and under different applied loads. The second set was on specimens with different fiber diameter and the same loading conditions. While crack arrest was observed in the first set, crack arrest was seen in the second set for the relatively large diameter fibers and specimen fracture for the relatively thin fibers. A method, based on fracture mechanics principles and crack opening displacements, for evaluating bridging tractions is outlined. Using this method, simulations for the bridging tractions and stress intensity factor were carried out using a linear crack opening profile. The total stress intensity factor was found to decrease with crack length. The debonding in the bridging zone, on specimens with different fiber spacing, was evaluated using a one dimensional debonding analysis. The model was calibrated with the debonding on the first fiber and consequently used to describe debonding on the bridging zone of specimens with different fiber spacing. In spite of the assumptions adopted in the present studies, the model seems to describe debonding well.     

Fatigue crack propagation from small holes in linear arrays

The propagation of fatigue cracks emanating from linear arrays of circular holes has been analytically investigated. An approximate fracture mechanics analysis is developed for plates containing cracked holes in rows which are either normal to or parallel to a uniaxial tensile load and the crack growth law % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaadsgaieGaca % WFHbacbaGaa43laiaa+rgacaWFUbGaa8xpaiaa-feacaGFOaGaeuiL % dqKaa83saiaa+LcadaahaaWcbeqaaiaa-jeaaaaaaa!41B0! \[ da/dn = A(\Delta K)^B \] is integrated over the K field of each array from a crack nucleus size to the critical crack length. Propagation lifetimes thus obtained are found to decrease with decreasing hole spacing for collinear arrays normal to the loading axis but to increase with decreasing spacing for holes aligned parallel to the stress axis. In addition, the absolute hole size, applied stress range, and the materials crack growth law are shown to affect the lifetimes of the various arrays.

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Zusammenfassung Es wird eine analytische Studie vorgelegt über die Fortpflanzung von Alterungsrissen, welche aus einer Anordnung von Rundlöchern herstammen. Für Platten mit gerissenen Löchern, welche in zu einer uniaxialen Spannung senkrecht oder parallel liegenden Reihen angeordnet sind, wurde eine auf die Bruchmechanik aufbauende angenäherte Analyse entwickelt. Das Gesetz der Rißfortpflanzung % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaadsgaieGaca % WFHbacbaGaa43laiaa+rgacaWFUbGaa8xpaiaa-feacaGFOaGaeuiL % dqKaa83saiaa+LcadaahaaWcbeqaaiaa-jeaaaaaaa!41B0! \[ da/dn = A(\Delta K)^B \]wurde \:uber die Kennzahl K der verschiedenen Arten von Anordnungen integriert und dies im Bereich von den Anfangsabmessungen des Risses bis zur kritischen L\:ange.Die sich aus dieser Berechnung ergebende Lebensdauern bei Rißfortpflanzung zeigen eine Verminderung für senkrecht zur Beanspruchungsaxe liegende kolineare Anordnung wenn man den Lochabstand verkleinert.Außerdem wird gezeigt, daß die für verschiedene Anordnungen ermittelte Lebensdauer von der absoluten Abmessung der Löcher, der Amplitude der angelegten Spannungen und dem dem Werkstoff eigenen Rißfortpflanzungsgesetz beeinflußt wird.

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Résumé On a étudié de manière analytique la propagation des fissures de fatigue provenant d'arrangements de trous circulaires. Pour des plaques comportant des trous fissurés en rangées orientées normalement ou parallèlement à une tension uniaxiale, on a développé une analyse approximative basée sur la mécanique de la rupture; la loi de propagation des fissures % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaadsgaieGaca % WFHbacbaGaa43laiaa+rgacaWFUbGaa8xpaiaa-feacaGFOaGaeuiL % dqKaa83saiaa+LcadaahaaWcbeqaaiaa-jeaaaaaaa!41B0! \[ da/dn = A(\Delta K)^B \] a \'et\'e int\'egr\'ee sur la caract\'eristique K de chaque type d'arrangements, entre la dimension de la fissure initiale et longueur critique.Les durées de vie en propagation déduite de cette opération font état d'une réduction lorsque diminue la distance entre trous pour des arrangements colinéaires normaux par rapport à l'axe de sollicitation.En outre, on montre que la durée de vie correspondant à divers arrangements de trous est affectée par la dimension absolue des trous, l'amplitude des contraintes appliquées, et la loi de propagation des fissures propre au matériau.

     

A critical analysis of the relationship between the energy release rate and the stress intensity factors for non-coplanar crack extension under combined mode loading

A critical analysis was made on the relationship between the energy release rate G and the stress intensity factors for non-coplaner crack extension under combined Mode I, II and III loading. Developing a method different from the application of Bueckner's equation, the equation of G was derived as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2% da9iaacUfacaGGOaGaeuOUdSMaey4kaSIaaGymaiaacMcacaGGVaGa% aGioaiabfY7aTjaac2facaGGOaGabm4sayaaiaWaa0baaSqaaiaabM% eacaWGIbaabaGaaGOmaaaakiabgUcaRiqadUeagaacamaaDaaaleaa% caqGjbGaaeysaiaadkgaaeaacaaIYaaaaOGaaiykaiabgUcaRiaacI% cacaaIXaGaai4laiaaikdacqqH8oqBcaGGPaGaey4kaSIabm4sayaa% iaWaa0baaSqaaiaabMeacaqGjbGaaeysaiaadkgaaeaacaaIYaaaaa% aa!5988!\[G = [(\kappa + 1)/8\mu ](\tilde K_{{\text{I}}b}^2 + \tilde K_{{\text{II}}b}^2 ) + (1/2\mu ) + \tilde K_{{\text{III}}b}^2 \], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciqa-Teaga% acamaaBaaaleaaieaacaGFjbGaa4NyaaqabaGccaqGGaGab83sayaa% iaWaaSbaaSqaaiaa+LeacaGFjbGaa4NyaiaabccaaeqaaKaaajaa+f% gacaGFUbGaa4hzaOGaa4hiaiqa-TeagaacamaaBaaaleaacaGFjbGa% a4xsaiaa+LeacaGFIbGaaeiiaaqabaaaaa!477C!\[\tilde K_{Ib} {\text{ }}\tilde K_{IIb{\text{ }}} and \tilde K_{IIIb{\text{ }}} \] are the stress intensity factors at the tip of an infinitesimal kink as formed by non-coplaner crack extension. Among the three existing equations of G which are mutually contradictory, Nuismer's equation and Wang's equation disagree with the present result. In the case where Mode III loading is not involved, Hussain et al's equation agrees with the present result, although their analysis seems to contain questionable points.

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Résumé On a procédé à une analyse critique de la relation entre la vitesse de relaxation de l'énergie G et les facteurs d'intensité de contrainte dans le cas de l'extension d'une fissure non coplanaire sous des modes de chargement I, II et III. En développant une méthode différente de l'application de l'équation de Bueckner, on a obtenu une équation pour G fonction des facteurs d'intensité de contrainte à l'extrémité d'un ressaut infinitésimal tel que formé par l'extension d'une fissure non coplanaire. Parmi les trois équations de G existantes qui se trouvent être mutuellement contradictoires l'équation de Nuismer et l'équation de Wang ne sont pas en accord avec les résultats obtenus dans le mémoire. Dans le cas où un mode de sollicitation III n'est pas pris en considération, l'équation de Hussain et al. est en accord avec les résultats obtenus bien que leur analyse semble contenir des points discutables.

     

Observation of deformation and damage at the tip of cracks in adhesive bonds loaded in shear and assessment of a criterion for fracture

The evolution of damage at the tip of cracks in adhesive bonds deforming in shear was monitored in real time using a high-magnification video camera. Brittle and a ductile epoxy resins were evaluated, with the bond thickness t being an experimental variable. An extensive zone of plastic deformation developed ahead of the crack tip prior to fracture. In the case of the brittle adhesive, for relatively thick bonds tensile microcracks formed within that zone. Increased loading caused the microcracks to grow from the interlayer to the interface, which led to a complete bond separation after interface cracks emanating from adjacent microcracks linked. In contrast, for the ductile adhesive the crack always grew from the tip. Strain gradients tended to develop there when the bond thickness was large.The adhesive shear strain was determined from fine lines scratched on the specimen edge. For both adhesives, the average crack tip shear strain at crack propagation rapidly decreased with increasing t. This effect was attributed to the changing sensitivity of the bond to the presence of flaws; thicker bonds can accommodate larger microcracks or microvoids which cause greater stress concentration. For a given bond thickness, the critical crack tip shear strain agreed well with the ultimate shear strain of the unflawed adhesive previously determined using the napkin ring shear test [12]. This suggests that the ultimate shear strain is a key material property controlling crack growth. The critical distortional strain energy/unit area of the unflawed adhesive W _s was determined from the area under the stress-strain curve in the napkin ring test. Good agreement between W _s and the adhesive mode II fracture energy was found for all joints tested except for relatively thick bonds. For the particular case of an elastic-perfectly plastic adhesive, the agreement above implies % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Deada% WgaaWcbaacbaGaa4xsaiaa+LeacaGFdbaabeaakiabg2da9iaa-Dfa% daWgaaWcbaGaa83CaaqabaGccqGHHjIUcaWF0bGaeqiXdq3aaSbaaS% qaaiaa-LhaaeqaaOGaeq4SdC2aaSbaaSqaaiaa-zgaaeqaaaaa!463A!\[G_{IIC} = W_s \equiv t\tau _y \gamma _f \].     

Crack layer analysis of fatigue crack propagation in rubber compounds

A methodology to characterize the resistance of rubber compounds to crack propagation (fracture toughness) is presented. A constitutive model based on the crack layer theory is utilized to extract the specific energy of damage ^*, a material parameter characteristic of the material's resistance to crack propagation and the dissipative characteristic, . The model expresses the rate of crack propagation as% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaalaaabaGaam% izaGqaciaa-fgaaeaacaWGKbGaa8Ntaaaaaaa!3AFA!\[\frac{{da}}{{dN}}\]= % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaeq% OSdiMaamOsamaaDaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiaadMha% caGGQaGaamOuamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadQeada% WgaaWcbaGaaGymaaqabaaaaaaa!41A5!\[\frac{{\beta J_1^2 }}{{y*R_1 - J_1 }}\]where da/dN is the cyclic rate of fatigue crack propagation (FCP), J ₁ is the energy release rate (tearing energy) and R ₁ is the resistance moment which accounts for the amount of damage associated with the crack advance. Microscopic examination revealed that crack tip microcracking is the dominant damage mechanism. Hence, R ₁ was evaluated as the area (m²) of microcracking surfaces per unit crack advance.Fatigue crack propagation data for a particular rubber compound have been analyzed using the present model. The proposed equation describes the entire FCP history in the compound. According to this model, ^* and for the compound investigated, are found to be 9.3 kJ m^-2 and 9.7×10^-9 m⁴/J-cycle, respectively.

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Résumé On présente une méthodologie pour caractériser la résistance de composés de caoutchouc à la propagation des fissures du point de vue de la ténacité à la rupture. Un modèle constitutif basé sur la théorie de la couche de fissuration est utilisé pour obtenir l'énergie spécifique d'endommagement ^*, un paramètre du matériau représentatif de sa résistance à la propagation d'une fissure, et une caractéristique de dissipation . Le modèle exprime la vitesse de propagation d'une fissure de fatigue par cycle da/dN en fonction de ces deux paramètres, de la vitesse de relaxation de l'énergie de cisaillement J ₁, et du moment résistif R ₁ qui tient compte de état de l'endommagement associé à la progression de la fissure. Un examen microscopique révèle que la microfissuration à l'extrémité de la fissure est le mécanisme déterminant de l'endommagement. Dès lors, on évalue R ₁ en fonction de l'aire de microfissuration (en m²) par unité de progression de la fissure.Des données de propagation de fissure de fatigue sont analysées à l'aide du présent modèle pour un composé de caoutchouc particulier. L'équation proposée décrit l'entièreté de la propagation de la fissure dans le composé. Des valeurs numériques pour ^* et pour de respectivement 9,3 kJ m^-2 et 9,7×10^-9 m^-4/J-cycle sont trouvées.

     

Contribution of bending energy losses to the apparent tear energy

When a strip is torn, energy is expended both in tearing it and in propagating a bend along each torn section. Estimates are given of the contribution of bending energy losses to the apparent tear energy. Experiments with highly-dissipative semi-crystalline polymers, torn with controlled amounts of bending, are then described. The bending energy losses ranged from 5 to 70 percent of the total tear energy, depending upon the degree of bending imposed, the thickness of the strip, and the extent to which it had been partly cut through before tearing. These results were in satisfactory agreement with approximate theoretical estimates. When the torn strips were allowed to take up naturally bent configurations under the action of the tearing force, then the contribution of bending energy losses to the apparent tear energy became rather independent of the strip dimensions and depended principally upon the dissipative nature of the material, represented by the fraction H of deformation energy that is not recovered. A general relationship is proposed between the apparent (G _c) and true (G _c) tear energies in this case: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Deada% qhaaWcbaGaa83yaaqaaiaa-DcaaaGccqGH9aqpcaWFhbWaaSbaaSqa% aiaa-ngaaeqaaOGaai4laiaacIcacaaIXaGaeyOeI0Iaa8hsaGqaai% aa+Lcaaaa!415E!\[G_c^' = G_c /(1 - H)\]. Values of H for the materials examined ranged from 30 to 70 percent. Thus, bending energy losses are expected to increase the tear energy by a factor of 1.4 × to 3.3 × for unconstrained tearing of these semi-crystalline polymers. Somewhat smaller increases were actually observed, ranging from 1.1× to 2×.

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Résumé Lorsque l'on déchire une bande, l'énergie est dépensée à la fois dans le déchirement et dans la propagation d'une flexion sur chacun des bords de la déchirure.On donne des estimations de la contribution des pertes d'énergie associées à ces flexions, à l'énergie apparente de déchirement. On décrit ensuite des expériences de déchirement de bandes en polymères semi-cristallins à haute dissipation, sous des conditions de flexion contrôlées. On a établi que la dissipation d'énergie associées à la flexion vaut de 5 à 70% de l'énergie totale de déchirement, selon le degré de flexion imposé, l'épaisseur de la bande, et la longueur de la coupe réalisée avant déchirure. Ces résultats ont été trouvés en accord satisfaisant avec les estimations théoriques. Lorsque les portions déchirées adoptent la configuration de flexion qui correspond à une situation naturelle sous l'effet des forces de déchirement, les pertes dues à l'énergie de flexion contribuent à l'énergie apparente de déchirement de manière relativement indépendante des dimensions de la bande, mais principalement dépendante de la nature dissipatoire du matériau, représentée par la fraction H de l'énergie de déformation non récupérée.Une relation générale est proposée dans ce cas entre l'énergie apparente (G _c) et réelle (G _c) de déchirement: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Deada% qhaaWcbaGaa83yaaqaaiaa-DcaaaGccqGH9aqpcaWFhbWaaSbaaSqa% aiaa-ngaaeqaaOGaai4laiaacIcacaaIXaGaeyOeI0Iaa8hsaGqaai% aa+Lcaaaa!415E!\[G_c^' = G_c /(1 - H)\]Les valeurs de H varieront de 0,3 à 0,7 selon les matériaux examinés. Dès lors, on s'attend à ce que les pertes par énergie de flexion accroissent l'énergie de déchirement d'un facteur 4,4× à 3,3× dans le cas d'un déchirement sans rétreint des polymères semi-cristallins étudiés.Dans la réalité, on a observé des accroissements un peu plus faibles, variant de 1,1× à 2×.

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Department of Aerospace Engineering and Mechanics, University of Minnesota     

An analysis for crack layer stability

Fracture in many materials propagates as a crack preceded by a zone of structural transformations (damage). The crack layer (CL) theory derives the law of propagation of crack and the surrounding damage zone within the framework of irreversible thermodynamics. Employing the Prigogine-Glansdorff criterion of evolution, the theory derives the stability conditions for CL propagation. In this paper the problem of uncontrolled crack propagation and crack arrest is addressed from the viewpoint of CL translational stability. CL propagation is controlled by the difference between J ₁, the energy release rate, and ^*R₁, the amount of energy required for material transformation. As a result, the necessary and sufficient conditions for CL instability are % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadQeadaWgaaWcbaGaaGymaiabgkHiTiabeo7aNnaaCaaameqa% baGaaiOkaaaaaSqabaGccaWGsbWaaSbaaSqaaiaaigdaaeqaamrr1n% gBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfmGccqWFLjsHcaaI% Waaaaa!4F23!\[J_{1 - \gamma ^* } R_1 \geqslant 0\]and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaeyOaIylabaGaeyOaIyRaamiBaaaadaqadiqaaiaa% dQeadaWgaaWcbaGaaGymaaqabaGccqGHsislcqaHZoWzdaahaaWcbe% qaaiaacQcaaaGccaWGsbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGa% ayzkaaGaeyOpa4JaaGimaaaa!4A4A!\[\frac{\partial }{{\partial l}}\left( {J_1 - \gamma ^* R_1 } \right) > 0\]respectively. CL propagation in polystyrene, a model material, is studied under remotely applied, fixed load fatigue (Case I) and under fixed displacement (Case II). For Case I, the sufficient condition of instability is met before the necessary condition, so that the latter becomes the controller of stability. For case II, neither sufficient nor necessary conditions of instability are satisfied. Hence, CL propagation remains stable, eventually resulting in crack arrest.

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Résumé Dans de nombreux matériaux, la rupture se propage sous la forme d'une fissure précédée d'une zone où s'effectuent des transformations structurales, qualifiées d'endommagement. La théorie de la couche fissurée (C.F) résulte de la loi de propagation d'une fissure et de l'existence de la zone fissurée environnante, dans le cadre d'une évolution thermodynamique irréversible.En recourant au critière d'évolution de Prigogine-Glansdorff, on déduit par la théorie les conditions de stabilité qui régissent la propagation de la C.F. Dans la présente étude, on se concentre sur le problème de la propagation incontrôlée et sur l'arrêt de la fissure du point de vue de la stabilite'e d'une C.F en translation. La propagation de la C.F est régie par la différence entre la vitesse de relaxation de l'énergie J ₁et la quantité d'énergie nécessaire pour provoquer une transformation du matériau j ^*R₁. Le résultat conduit à exprimer les conditions nécessaires et suffisantes de l'instabilité de la C.F suivant respectivement % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadQeadaWgaaWcbaGaaGymaiabgkHiTiabeo7aNnaaCaaameqa% baGaaiOkaaaaaSqabaGccaWGsbWaaSbaaSqaaiaaigdaaeqaamrr1n% gBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfmGccqWFLjsHcaaI% Waaaaa!4F23!\[J_{1 - \gamma ^* } R_1 \geqslant 0\]d'une part et % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaeyOaIylabaGaeyOaIyRaamiBaaaadaqadiqaaiaa% dQeadaWgaaWcbaGaaGymaaqabaGccqGHsislcqaHZoWzdaahaaWcbe% qaaiaacQcaaaGccaWGsbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGa% ayzkaaGaeyOpa4JaaGimaaaa!4A4A!\[\frac{\partial }{{\partial l}}\left( {J_1 - \gamma ^* R_1 } \right) > 0\]d'autre part.On étudie la propagation de la C.F dans un matériau pour maquettes, le polystyrène, sous une charge de fatigue appliquée à charge constante (Cas I) ou à déplacements constants (Cas II). Pour le Cas I, la condition suffisante d'instabilité est satisfaite avant la condition nécessaire, qui de ce fait contrôle de la stabilité. Pour le Cas II, aucune des conditions n'est satisfaite, de sorte que la propagation de la C.F demeure stable et conduit en conséquence à un arrêt de fissuration.

     

Strength characteristics and fatigue crack growth in a composite with long aligned fibers

Strength values and fatigue crack growth on a specially made composite material are reported. The composite specimen consisted of an epoxy matrix and one layer of long aligned glass fibers that were equally spaced. The results on strength showed that for a range of fiber spacing , the composite's strength _c, scaled with a fiber spacing in the form of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa!39C2!\[\sigma \] c % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca% qG7oaaleqaaaaa!395B!\[\sqrt {\text{\lambda }} \] = % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOUdaaa!393F!\[{\text{\kappa }}\]. Based on dimensional arguments the constant was found proportional to the fracture toughness of the matrix. Fatigue crack propagation experiments were performed on specimens with different fiber spacing and applied loads. The crack speed reached a steady mode of propagation in specimens where relation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa!39C2!\[\sigma \] c % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca% qG7oaaleqaaaaa!395B!\[\sqrt {\text{\lambda }} \] = % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOUdaaa!393F!\[{\text{\kappa }}\] was satisfied. The same mode of propagation was reached for the debonding along fibers in the bridging zone as well as the crack opening displacement. The crack opening displacement at a fiber location and the corresponding debonding were linearly related. Within the resolution of the observations, no fiber fracture was seen in the bridging zone. Using a standard Green's function, stress intensity factor simulations were carried out for different types of tractions on the fibers in the bridging zone. When the fibers in the bridging zone were under a uniform load, the total stress intensity factor K _t, at the crack tip, was found constant at the steady state and proportional to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa!39C2!\[\sigma \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIu6aaO% aaaeaacqaH7oaBaSqabaaaaa!3B3F!\[\infty \sqrt \lambda \]. Assuming that K _t is constant during steady crack growth, the results of the simulations were used to correlate steady crack speed in three sets of data. Dimensional analysis of the steady crack speed was carried out as an attempt to identify important parameters and the role of the fiber spacing in the fracture of the composite specimens. The steady crack speeds were correlated with the total stress intensity factor for each fiber spacing. The resulting exponents were found to be about 20 percent different. Assuming that at steady state the energy release rate for an interfacial crack is proportional to t t ² r, where t is the stress carried by a fiber [20], a power expression for the rate of debonding with t % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq% aHepaDaSqabaaaaa!39DF!\[\sqrt \tau \] was found to have an exponent approximately equal to that for the steady crack speed.     

On asymmetry in Stewartson layers

There are two basic conclusions reached in this paper. It is shown first that the structure of free shear layers in a rotating fluid change only slightly when an asymmetry is introduced into the geometry; in fact, once a Poisson equation which describes the flow in the interior Taylor column is solved, the behaviour in both % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaG4maaaaaaa!3775!\[\frac{1}{3}\] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3776!\[\frac{1}{4}\]-layers is understood for all situations. The second result derived is that when the Stewartson layers are along solid sidewalls in an asymmetric configuration, then the velocity within the interior has additional components induced by the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3776!\[\frac{1}{4}\]-layer; the first perturbation is O(E^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3776!\[\frac{1}{4}\]}).     

Near tip fields for a stationary mode III crack between a linear elastic and an elastic power law hardening material

The asymptotic stress field near the tip of an antiplane crack lying along a planar bimaterial interface between an elastic and an elastic power law hardening material is analysed. Deformation plasticity theory is assumed in the analysis. We show that the shear stress field near the tip is of the form% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabes8a0naaDaaaleaacaWGPbaabaGaeyySaelaaOGaaeiiaiab% gYJi+jaabccacaWGYbWaaWbaaSqabeaadaWcgaqaaiabgkHiTiaaig% daaeaacaGGOaGaamOBaiabgUcaRiaaigdacaGGPaaaaaaakiaabcca% caWGubWaa0baaSqaaiaaicdadaWgaaadbaGaeyySaelabeaaaSqaai% aacIcacaWGPbGaaiykaaaakiaabccacaGGOaGaeqiUdeNaaiykaiaa% bccacqGHRaWkcaqGGaGaamOCamaaCaaaleqabaGaamiDamaaBaaame% aacaaIXaaabeaaliabgkHiTiaaigdaaaGccaqGGaGaamivamaaDaaa% leaacaaIXaWaaSbaaWqaaiabgglaXcqabaaaleaacaGGOaGaamyAai% aacMcaaaGccaqGGaGaaiikaiabeI7aXjaacMcacaqGGaGaey4kaSIa% aeiiaiabl+UimjaabccacqGHRaWkcaqGGaGaamOCamaaCaaaleqaba% GaamiDamaaBaaameaacaWGRbaabeaaliabgkHiTiaaigdaaaGccaqG% GaGaamivamaaDaaaleaacaWGRbWaaSbaaWqaaiabgglaXcqabaaale% aacaGGOaGaamyAaiaacMcaaaGccaqGGaGaaiikaiabeI7aXjaacMca% caqGGaGaey4kaSIaaeiiaiabl+Uimbaa!809A!\[\tau _i^ \pm {\text{ }} \sim {\text{ }}r^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {(n + 1)}}} \right. \kern-\nulldelimiterspace} {(n + 1)}}} {\text{ }}T_{0_ \pm }^{(i)} {\text{ }}(\theta ){\text{ }} + {\text{ }}r^{t_1 - 1} {\text{ }}T_{1_ \pm }^{(i)} {\text{ }}(\theta ){\text{ }} + {\text{ }} \cdots {\text{ }} + {\text{ }}r^{t_k - 1} {\text{ }}T_{k_ \pm }^{(i)} {\text{ }}(\theta ){\text{ }} + {\text{ }} \cdots \]for. Here r is the radial distance from the crack tip, is the angle measured from the interface, n is the hardening exponent, and + and — indicate the plastic and elastic regions respectively. The exponents t _k are uniquely determined by n, and for k1,t _k+1> t _k , t ₁. For kM, where M is the largest positive integer for which (n(M+1)-M)/(n+1) < 0.5 (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaakaaabaGaamOBamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaa% iodacaaI0aGaamOBaiabgUcaRiaaigdaaSqabaaaaa!431D!\[\sqrt {n^2 + 34n + 1} \] + 1 + n – 1)/(n + 1),t _k = (n(k + 1) – k)/(n + 1). The corresponding angular functions % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadsfadaqhaaWcbaGaam4AamaaBaaameaacqGHXcqSaeqaaaWc% baGaaiikaiaadMgacaGGPaaaaOGaaiikaiabeI7aXjaacMcaaaa!45AB!\[T_{k_ \pm }^{(i)} (\theta )\] are determined by the J-integral and material parameters and can be obtained completely from the asymptotic analysis. Some of the terms of stresses with kM may be singular. For k>M, t _k can be obtained numerically, and the corresponding % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadsfadaqhaaWcbaGaam4AamaaBaaameaacqGHXcqSaeqaaaWc% baGaaiikaiaadMgacaGGPaaaaOGaaiikaiabeI7aXjaacMcaaaa!45AB!\[T_{k_ \pm }^{(i)} (\theta )\] can be obtained completely or within multiplicative constants. All the terms of stresses with k>M vanish as r , when r0, where >0, for all 1<n<. It is important to note that although the individual terms of the stress expansion is variable separable, the resultant stress field is non-separable. The values of t ₁,...,t ₅ for 1<n20 and the first three terms of stresses for various values of n and material parameters are computed explicitly in the paper. Our analysis shows that, in the series solution for stresses in the plastic domain, the effect of the linear elastic material appears in the second or higher order terms depending on the value of n. In spite of this effect of elasticity on the higher order terms, the region of dominance of the HRR field in the plastic zone % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadkhadaahaaWcbeqaaiabgkHiTmaalyaabaGaaiikaiaaigda% aeaacaWGUbGaey4kaSIaaGymaiaacMcaaaaaaOGaamivamaaDaaale% aacaaIWaWaaSbaaWqaaiabgUcaRaqabaaaleaacaGGOaGaamyAaiaa% cMcaaaGccaGGOaGaeqiUdeNaaiykaaaa!4B3E!\[r^{ - {{(1} \mathord{\left/ {\vphantom {{(1} {n + 1)}}} \right. \kern-\nulldelimiterspace} {n + 1)}}} T_{0_ + }^{(i)} (\theta )\] may be significantly reduced compared to the corresponding region of dominance when the crack is in a homogeneous elastic power law hardening material.     

Application of the Finite Element Iterative Method to cracks and sharp notches in orthotropic media

In this paper, the asymptotic singular fields at cracks and sharp notches in isotropic and orthotropic media are numerically investigated. The Finite Element Iterative Method (FEIM) is used for evaluating the power of the singular field at the notch root under three different loading conditions: tension, shear, and a combination of tension and shear. The numerical results for the isotropic case, obtained by the FEIM, are in excellent agreement with the results from analytical solutions. In tension, the singularity at the notch root converges to the strong singularity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS% baaSqaaiaadggaaeqaaaaa!38B3!\[\lambda _a \], while under shear, convergence is to the weak singularity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS% baaSqaaiaadkgaaeqaaaaa!38B4!\[\lambda _b \]. When the loading is a combination of tension and shear, the singularity at the notch root always converges to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS% baaSqaaiaadggaaeqaaaaa!38B3!\[\lambda _a \] The stress singularity results for orthotropic materials followed the same trend. However, the stress singularities % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS% baaSqaaiaadggaaeqaaaaa!38B3!\[\lambda _a \] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS% baaSqaaiaadkgaaeqaaaaa!38B4!\[\lambda _b \] are strongly dependent on the elastic properties of the material. Under combined tension and shear loading, the number of iterations for convergence significantly increases, as the loading changes from tension-dominated to shear-dominated. This study has demonstrated the usefulness of the FEIM for the evaluation of asymptotic singular fields at sharp notches in orthotropic materials.     

Application of load separation and normalization methods for polycarbonate materials

Ductile fracture methods recently developed for metallic materials are applied here for the first time to polymeric materials. These include the load separation method for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS% baaSqaaiaabchacaqGSbaabeaaaaa!39A7!\[\eta _{{\text{pl}}} \]determination and the normalization method for J-R curve calculation. A polycarbonate is chosen for this investigation. A set of blunt notched compact specimens were tested to evaluate load separation. In addition precracked specimens with different thicknesses were prepared for comparing the standard multiple specimen test method with the single specimen normalization method.The results show that these new methods are applicable to ductile polymer materials. The % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS% baaSqaaiaabchacaqGSbaabeaaaaa!39A7!\[\eta _{{\text{pl}}} \]value for the polycarbonate compact specimens is approximately the same as the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS% baaSqaaiaabchacaqGSbaabeaaaaa!39A7!\[\eta _{{\text{pl}}} \]for metallic compact specimens. This means that material deformation properties like yield stress and work hardening have little influence on the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS% baaSqaaiaabchacaqGSbaabeaaaaa!39A7!\[\eta _{{\text{pl}}} \]values of compact type specimens. The normalization method gives a J-R curve from a single specimen which is comparable to the J-R curves from the standard multiple specimen testing method.     

Planar textures in superfluid 3He-A

Making use of the Ginzburg-Landau free energy, singularities such as disgyrations and vortex lines and stable textures in the planar (or two-dimensional) texture are studied. Here both % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaieqaceWFSbGbaK% aaaaa!3883!\[{\hat l}\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaieqaceWFKbGbaK% aaaaa!387B!\[{\hat d}\] vectors, characterizing the texture in ³He-A, are assumed parallel to each other and lie always in a plane. The dynamics of disgyrations is also considered.Work supported by the National Science Foundation under grant number DMR76-21032.