J-integral correlation of the initiation of slow crack growth in linear polyethylene

In previous investigations it has been shown that the regime of slow crack growth and the time to failure depend directly on the initial rate of crack-tip opening and damage. In this paper we demonstrate that a more appropriate fracture parameter to correlate various load levels, crack lengths, and specimen geometries under small-scale-yielding conditions is the J-integral rather than K. Since the initial rate of damage depends directly on the J-integral, the phenomenon of long-time failure by slow crack growth should be correlated by the J-integral for linear PE.

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Résumé Au cours d'études précédentes, on a montré que le régime de croissance lente d'une fissure et le temps nécessaire à la rupture dépendent directement de la vitesse initiale de l'ouverture et de l'endommagement à l'extrémité de la fissure. Dans ce travail, on démontre que l'intégrale J est un meilleur paramètre de rupture que K pour mettre en corrélation différentes valeurs de la charge, longueurs des fissurs et géométries d'éprouvettes, sous des conditions d'écoulement plastique à faible échelle. Comme la vitesse d'endommagement initiale dépend directement de l'intégrale J, on devrait pouvoir corréler avec J le phénomène de rupture différée sur une longue durée, résultant d'une croissance lente de fissure, dans le cas d'un polyéthyléne linéaire.

     

The relationship of the initiation stage to the rate of slow crack growth in linear polyethylene

The rate of notch opening was observed in single-edge notched tension specimens of linear high-density polyethylene under plane strain conditions as a function of applied stress and notch depth. The initial rate of notch opening was 22K ^4.3 m min^–1 (where units ofK are MN m^–3/2). The initial rate of notch opening was constant until the opening at its root was 25 to 30m for all stresses and notch depths. The accelerating part of the notch opening against time curve and the time to failure could be predicted from the initial rate of notch opening. It is concluded that the same mechanism governs the initiation stage and the subsequent crack growth rate.Visiting Scholar from the University of Science and Technology of China, Hefei, China.     

The initiation of slow crack growth in linear polyethylene under single edge notch tension and plane strain

The kinetics and microstructural changes associated with the initiation of slow crack growth in PE were measured. The initiation process consists of an instantaneous deformation zone which grows at a constant velocity until the beginning of fracture. The velocity of the damaged zone accelerates when the fibril fracture begins at the root of the initial notch. It was found that the initial velocity of the deformation zone depended on stress to about the 4th power and had an activation energy of about 100 kJ mol^–1; these results are about the same as those found by Chan and Williams for the crack growth velocity. It is concluded that both crack initiation and crack growth are governed by the same fundamental process, notably fibril thinning.     

The effect of molecular weight on slow crack growth in linear polyethylene homopolymers

The rate of initiation and growth of cracks in linear high-density polyethylene with different molecular weights was observed in single-edge-notched tensile specimens under plane strain condition as a function of applied stress, notch depth and temperature. The initial rates of crack initiation all have the form of C ^m a ₀ ⁿ exp (–Q/RT) or AK ^pexp (–Q/RT) where = stress, a ₀ = notch depth and K= stress intensity factor. For the different molecular weights, m, n, P and Q are almost the same where m=5, n=2, P=4.7 and Q=115 kJ mol^–1, but the constants C and A varied as (¯M _w–¯M _c)^–1 where ¯M_c is a limiting molecular weight for sudden fracture. A molecular model based on tie-molecules has been used to explain the dependence on ¯M _w. The effect of ¯M _w on the fast-fracture strength at low temperature and the relationship to tie-molecules have also been investigated. Quantitative relationships between the concentration of tie-molecules and the fracture behaviour have been obtained.     

Predicting failure from the initiation stage of slow crack growth in polyethylene

The crack opening displacement against time was measured in linear polyethylene as a function of stress, notch depth, and temperature for three-point bending under plane strain conditions. The experimental conditions were accurately controlled so that the scatter in the rate of damage was within ±20%. Microscopic observations showed that the shape of the damaged zone was triangular for stresses less than one-half the yield point and that the length was predictable from measurements of the crack opening displacement. The initial rate of damage prior to crack growth was constant up to crack opening displacement = 15 to 25Μm. The entire shape of the crack opening displacement-time curve and the time to complete failure could be predicted from the initial damage rate. A theory based on the observed microcrazing is presented which explains the dependence of damage rate on stress intensity and microstructural parameters.     

Mechanistic analysis of fatigue crack initiation in medium-density polyethylene

Kinetics parameters of craze evolution preceding fatigue crack initiation (FCI) in mediumdensity polyethylene (MDPE) pipe materials were determined and analysed within fracture mechanics theory. A single craze initially preceded the notch tip, a root craze, which subsequently became accompanied by a few side crazes. Crack initiation transpired after the craze-zone growth had reached its fully developed configuration. The length of the root craze of the fully developed zone was found to be equal to the length of the first discontinuous crack band on the fracture surface. The growth of the root-craze length and the crack-tip opening displacement (CTOD) followed a power law over the major portion (94%) of the FCI time. Measurable rupture of the craze material was only noted within the final portion of the FCI time and was associated with exponential increase of the CTOD. The Dugdale/Barenblatt model overestimated the craze length by 30% and underestimated the CTOD by 50% which was hypothesized to be due to multiple crazing at the notch tip.     

The fracture mechanics of slow crack growth in polyethylene

The following theoretical equation was obtained for the rate of initiation fx- of slow crack growth in polyethylene:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmqr1ngBPrgitLxBI9gBaerbd9wDYLwz% YbItLDharuavP1wzZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasa% acOqpw0le9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfGaqaaiabes7aKbWc% beqaaiaac6caaaGccqGH9aqpdaWcaaqaaiabeo8aZnaaBaaaleaaca% WG5baabeaacaWGlbWaaWbaaWqabeaacaaI0aaaaSGaaiikaiaaigda% cqGHsislcqaHZoWzdaahaaadbeqaaiaaikdaaaWccaGGPaWaaWbaaW% qabeaacaaIYaaaaaGcbaGaeq4TdGMaamizaiaadweadaahaaWcbeqa% aiaaikdaaaGccqaHdpWCdaqhaaWcbaGaam4yaaqaaiaaikdaaaaaaa% aa!5AB3!\[\mathop \delta \limits^. = \frac{{\sigma _y K^4 (1 - \gamma ^2 )^2 }}{{\eta dE^2 \sigma _c^2 }}\] where = applied stress, K = stress intensity, = Poisson's ratio, E = Young's modulus, _c = stress to produce a craze, _y = yield point, d = primordial thickness of the craze and = the intrinsic viscosity of the fibrils of the craze. The dependence of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmqr1ngBPrgitLxBI9gBaerbd9wDYLwz% YbItLDharuavP1wzZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasa% acOqpw0le9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfGaqaaiabes7aKbWc% beqaaiaac6caaaaaaa!4668!\[\mathop \delta \limits^. \] on K agrees with the experimental data. The experimental values of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmqr1ngBPrgitLxBI9gBaerbd9wDYLwz% YbItLDharuavP1wzZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasa% acOqpw0le9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfGaqaaiabes7aKbWc% beqaaiaac6caaaaaaa!4668!\[\mathop \delta \limits^. \] vary by a factor of 10⁷ depending on the type of polyethylene. This large variation in % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmqr1ngBPrgitLxBI9gBaerbd9wDYLwz% YbItLDharuavP1wzZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasa% acOqpw0le9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfGaqaaiabes7aKbWc% beqaaiaac6caaaaaaa!4668!\[\mathop \delta \limits^. \]% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmqr1ngBPrgitLxBI9gBaerbd9wDYLwz% YbItLDharuavP1wzZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasa% acOqpw0le9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfGaqaaiabes7aKbWc% beqaaiaac6caaaaaaa!4668!\[\mathop \delta \limits^. \] is directly related to intrinsic viscosity which evolved from the theory.     

The role of crazes in the crack growth of polyethylene

Brittle slow crack growth, or stress cracking, is a major concern in many applications of polyethylene materials. Using a constant tensile load test and removing specimens prior to complete failure, details of the crack tip region can be discerned in both butene and hexene copolymerized polyethylene. In both the presence and absence of an accelerating environment (Igepal CO-630), it was found that crazes formed at the crack tip, although secondary crazes were also evident in the specimens removed from the Igepal. Multiple crack arrest lines were clearly evident, suggesting a stick-slip mechanism under static load. The appearance of the craze zone at the crack tip can be explained through invoking an interlamellar failure model.     

Micromechanisms of crack initiation in thin films and thick sections of polyethylene

The morphology of the micro-deformation and fracture processes that occur in the neighbourhood of a notch prior to macroscopic crack growth were determined by a variety of microscopic techniques. The first micro-event that was observed in all instances was the formation of crack-like pores. The shape of the porous zone could not be deduced from existing yield criteria. In thick sections the porous zone transformed to a fibrillated region and in thin films the porous zone tends to transform into a continuous oriented zone. Fibril fracture consisted of a slow process of thinning by shear followed by a rapid rupture. The above micro-events precede macroscopic crack growth in polyethylene.     

The crack resistance in polybutylene terephthalate (PBT) at crack initiation and during steady crack growth

The resistance to crack initiation and quasi-static crack propagation is investigated for a polybutylene terephthalate (PBT) using annealed and unannealed three-point bend specimens. The resistance to crack initiation (R _i) is determined based on the generalized locus method which determines the resistance to crack growth including crack initiation utilizing the locus of characteristic points on the load against load-point displacement curves of specimens which differ only in initial crack length. This generalized locus method also enables us to investigate the invariance of the crack resistance value along the locus line. The steady state crack resistance (R _p) during quasi-static crack propagation is determined utilizing the functional relation between total essential energy (U _f) for complete fracture and the initial ligament length. The total essential energy is the sum of the blunting energy and the integration of the resistance to crack propagation with respect to the cracking area. The invariance of the crack initiation resistanceR _i, and the steady state resistanceR _p is discussed based on the experimental results of the annealed and unannealed specimens which show different sizes of crack tip plastic deformation.     

J-integral studies of crack initiation of a tough high density polyethylene

The J-integral is applied to characterise the fracture initiation of a tough high density polyethylene for which the concepts of linear elastic fracture mechanics (LEFM) are inapplicable for reasonably sized specimens due to extensive plasticity. The multiple specimen resistance curve technique recommended by the ASTM is the basic method employed. A formulation based on the finite difference in crack area between two otherwise identical specimens is used to determine J _Ic for comparison with the result obtained from the basic method. A comparison is also made with the results obtained from the Bilby, Cottrell & Swinden (BCS) model of yielding ahead of a crack.Tests are performed in the temperature range from +23°C to –80°C.

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Résumé On applique l'intégrale J à la caractérisation de l'amorçage d'une rupture dans un polyéthilène tenace à haute densité pour lequel les concepts de la théorie linéaire élastique de la mécanique de rupture sont inapplicables dans le cas d'éprouvette de dimension raisonnable, en raison d'une plasticité étendue. La méthode de base utilisée est la technique de la courbe de résistance d'une éprouvette multiple, telle que recommandée par l'ASTM. Une formulation basée sur les différences finies dans la zone de fissuration entre deux éprouvettes identiques par ailleurs, est utilisée à la détermination de J_Ic en vue d'une comparaison avec les résultats obtenus à partir de la méthode de base. Une comparaison est également effectuée avec les résultats obtenus par l'utilisation du modèle de Bilby, Cottrell et Swinden relatif à l'écoulement plastique en avant de la fissure.Les essais ont été effectués dans une gamme de températures comprise entre +23°C et –80°C.

     

Fatigue crack initiation and growth in solder alloys

In modern electronic packaging, especially surface mount technology (SMT), thermal strain is usually induced between components during processing, and in service, by a mismatch in the thermal expansion coefficients. Since solder has a low melting temperature and is softer than other components in electronic packaging, most of the cyclic stresses and strains take place in the solder. Fatigue crack initiation and fatigue crack propagation are likely to occur in the solder even when the cyclic stress is below the yield stress. It is an objective of this research to study the behaviour of fatigue crack initiation and propagation in both lead‐containing solder (63Sn‐37Pb), and lead‐free solders (Sn‐3.5Ag). The effect of alloying (Cu and Bi addition), frequency, tensile hold time and temperature on low cycle fatigue (LCF) behaviour of the solders is discussed. Mechanisms of LCF crack initiation and propagation are proposed and LCF life prediction, based on the various models, is carried out.     

Fatigue crack initiation and crystallographic crack growth in an austenitic stainless steel

The lifetime of a specimen under cyclic loading is usually limited by the initiation and growth of microcracks. Experimental results for the austenitic stainless steel X6 CrNiNb 18-10 are given and a model for crack initiation and crack growth in the first grains of a polycrystalline aggregate is proposed.     

The effect of repeating load on the crack growth initiation and crack propagation in delayed failure

The delayed failure test under repeating load was carried out with pre-cracked specimen. The incubation time and the crack propagation rate were correlated with the stress intensity factor K.

The incubation time is decreased by the superposition of repeating load, as the range of stress intensity factor ΔK or the repeating frequency f increase. The reason can be explained by the promotion of corrosion reaction due to, e.g. the destruction of oxide film on the crack tip, which facilitates the invasion of hydrogen atoms into the material.

The crack propagation rate da/dt is decreased by the superposition of repeating load, and there exist two valleys of crack propagation rate minima on the da/dt vs f and da/dt vs ΔK curves. One valley corresponds to the interaction between the cyclic movement of the region with tri-axial tensile stress and the hydrogen atoms diffused from crack tip, which disturbs the concentration of hydrogen atoms. Another seems te correspond to the generation of retained compressive stress which reduces the effective stress intensity at crack tip and supresses the invasion and diffusion of hydrogen atoms.     

Discontinuous crack growth in polyethylene under a constant load

The kinetics of slow crack growth were measured in a polyethylene copolymer in a notched tensile specimen under constant load. The microscopic changes in the crack morphology were linked to the crack opening displacement and to the crack advance. The jump distance during the discontinuous crack growth decreased as the applied stress decreased. The jump distance decreased as the temperature decreased because the yield point was increasing. These observations are explainable by the Dugdale theory. The initiation time for fracture depends on the rate of disentanglement of the fibrils in the craze and occurs in the fibrils adjacent to a tough skin at the base of the craze. The crack grows until it meets fibrils whose strength matches the value of the stress intensity at which point the crack is arrested. Re-initiation of fracture occurs when the fibrils at the root of the crack have been sufficiently weakened by the process of disentanglement. Thus, the overall kinetics depend on (1) the rate of disentanglement of the fibrils, (2) the gradient in the fibrilla strength between the base and the tip of the craze, (3) the value of the stress intensity, and (4) the yield point of the matrix.     

Application of the crack layer theory to modeling of slow crack growth in polyethylene

A crack and a domain of highly fibrillated and stretched material ahead of the crack (process zone), commonly observed in polyethylene, are considered as a system called the crack layer. Slow crack layer growth is assumed to be a result of interactions between the crack, process zone and the rest of the body, as well as of degradation of the process zone material. The energy balance for process zone formation and crack layer advance is presented. The equations governing crack layer propagation are formulated and numerically solved. The proposed mechanism of fracture process models the discontinuous crack growth often observed in polyethylene, and predicts the relationship between the crack growth rate and the stress intensity factor consistent with the experimental one. The dependence of the lifetime on load is discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

A theory of crack initiation and growth in viscoelastic media

The theory of crack growth developed in Parts I and II is used to predict crack velocity and failure time for an elastomer under simple uniaxial and biaxial stress states. Included are consideration of the effect of specimen size on failure time for initially small cracks, experimental determination of fracture properties, the effect of strain level on crack propagation, and the validity of the plane strain assumption. The second half of the paper examines the effect upon crack velocity of nonlinear viscoelasticity of failing material at the crack tip in media which is otherwise linearly viscoelastic. The dependence of fracture behavior on environmental changes and aging is also considered.

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Résumé On utilise la théorie de croissance de la fissure développée dans les parties I et II pour réduire la vitesse de fissuration et le temps de rupture d'un élastomére soumis à des états de contraintes uniaxiales et biaxiales. Le mémoire comporte des considérations sur l'effet de la dimension des éprouvettes sur le temps de rupture au départ de petites fissures, la détermination expérimentale des propriétés de la rupture, l'étude de l'effet du niveau de déformation sur la propagation des fissures, et de la validité d'hypothèses d'état plan de déformation.La seconde partie de l'article comporte l'examen de l'effet d'une viscoélasticité non linéaire dans la matière qui se rompt à la pointe d'une fissure, laquelle est par ailleurs sise dans un milieu caractérisé par une viscoélasticité linéaire.On considère également la manière dont le comportement à la rupture dépend du vieillissement et des changements de l'environnement.

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Parts I and II are published, respectively, in Vol. 11, No. 1 (1975) 141–159, and Vol. 11, No. 3 (1975) 369–388.     

A theory of crack initiation and growth in viscoelastic media

A theory is developed for predicting the time-dependent size and shape of cracks in linearly viscoelastic, isotropic media. First, the effect of a narrow zone of disintegrating material at the crack tip on opening displacement and on a finite stress distribution ahead of the tip is examined for elastic materials. Extension to viscoelastic media is then made. Although the undamaged portion of the continuum is assumed linear, no significant restrictions are placed on the nature of the zone of failing material at the crack tip and, therefore, this material may be highly nonlinear, rate-dependent, and even discontinuous. Finally, formulation of the problem is completed by introducing a local energy criterion of failure at the tip which is applicable to both constant and transient tip velocities. Parts II–IV, to appear in succeeding issues, will cover approximate methods of analysis and several applications of the theory.     

A theory of the initiation of creep crack growth

A computer simulation of the time dependent development of the plastic zone ahead of a crack loaded in uniform tension was performed. The material was assumed to deform according to a creep law relating the local strain rate to the local stress. The plastic zone was modelled by an array of edge dislocations coplanar with the crack. For a given time the stress was found to be uniform in a region ahead of the crack. This region increased and the local stress decreased with increasing time. The distribution of dislocations in the zone at a given time was found to be almost the same as that given by the Bilby, Cottrell and Swinden model (1963) if the friction stress in that model was replaced by an apparent friction stress equal to the uniform stress ahead of the crack. This apparent friction stress is dependent on both the applied stress and time. Assuming a critical crack opening displacement (COD) or a critical value of theJ integral,J _c, to be the criteria for the onset of the creep crack growth the initiation time can be calculated using the results of this study. A good agreement between the theory and experiment is obtained for two different CrMoV steels. This comparison with experiments suggests that the COD is an appropriate crack growth initiation parameter for both ductile and brittle materials whilstJ _cdoes not seem to be applicable in creep fracture.

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Résumé Une simulation par calculateur du développement fonction du temps de la zône plastique située devant une fissure soumise à tension uniforme a été effectuée. Le matériau est supposé pouvoir se déformer suivant une loi de fluage mettant en relation la vitesse de déformation locale et la tension locale. La zône plastique est représentée par une série de dislocations-coin coplanaires à la fissure. Pour un temps déterminé, la tension a été trouvée uniforme dans la région située de vant la fissure. Cette région s'étend, et la tension locale décroit, lorsque le temps s'accroit. La distribution des dislocations dans la zône à un moment déterminé est trouvée être sensiblement la même que celle donnée par le modèle de Bilby, Cottrell et Swinden (1963), pour autant que la tension de friction dans ce modèle soit remplacée par une tension de friction apparente égale à la tension uniforme située en avant de la fissure. Cette tension apparente de friction dépend à la fois de la contrainte appliquée et du temps. En supposant qu'une valeur critique du COD ou de l'intégraleJ,J _cconstitue le critère pour le démarrage d'une fissure de fluage, la durée de l'amorçage peut être calculée en utilisant les résultats de cette étude. Un bon accord entre la théorie et l'expérience a été obtenu dans le cas de deux aciers au CrMoV différents. Cette comparaison avec l'expérience suggère que le COD est un paramètre valable pour l'accroissement d'une fissure dans le cas de matériaux ductiles et de matériaux fragiles, tandis queJ _cne paraît pas applicable dans le cas des ruptures dues au fluage.