Plastic deformation in a thick transversely isotropic layer containing a penny-shaped crack

Summary The width of a thin plastic annular zone formed during the deformation of a pennyshaped crack in a transversely isotropic layer of an ideal elasto-plastic material is determined. Considered are the cases where the penny-shaped crack is extended by normal stresses and by torsional stresses. The faces of the layer are shear free and deformation of the plastic zone around the penny-shaped crack occurs according to the Dugdale hypothesis. For each case, the solution of the problem is reduced to a Fredholm integral equation of the second kind. Iterative solutions are obtained for small values of the parameters and numerical results for the width of the plastic zone are determined. Graphical results showing the effect of transverse isotropy upon the width of the plastic zone are also presented.With 6 Figures     

The fatigue growth of a disc-shaped crack

This paper presents a simple model appropriate for discussing the fatigue growth of a disc-shaped crack in an infinity body. It is found that the crack growth rate is proportional to the fourth power of the stress intensity factor.

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Zusammenfassung Diese Arbeit präsentiert ein einfaches Modell, welches geeignet ist, die Ermuedungsentwicklung eines diskfoermigen Risses in einem unendlichen Koerper zu diskutieren. Es wurde festgestellt, dass die Risszunahmsrate proportional zu der vierten Potenz des Druckintensitaetsfaktors ist.

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Résumé Le mémoire présente un modèle simple approprié à la discussion de 1'extension par fatigue d'une fissure de forme discale dans tin corps infini.On trouve que la vitesse d'extension de la fissure est proportionnelle à la quatrième puissance du facteur de concentration des tensions.

     

Plastic zone formation around an arresting crack

We have followed the evolution of strain near the tip of an arresting cleavage crack in a pressure vessel steel using a strain gage technique. The measurements were made using a linear array of strain gages located along the intended crack path and 0.65B (B is the specimen thickness) above this plane. the gages were read using instrumentation capable of resolving strain in time intervals of between 2 and 4 s. The results were obtained in a thick plate, large enough that reflected elastic waves did not contribute significantly to the strain records or crack motion during the time interval of interest.To within the resolution of this technique, the gages did not detect plasticity near the rapidly propagating cleavage crack tip. At the instant of arrest, however, the strain gages detected a plastic zone which increased in intensity over a period of several milliseconds. We have interpreted this increasing intensity as an elastic-plastic boundary emanating from the arresting crack tip and eventually reaching an equilibrium size, i.e., that predicted from the applied stress intensity factor and the static plastic properties. A simple analysis is presented to estimate the velocity of this elastic-plastic boundary from the strain data and results are given for several experiments.

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Résumé En utilisant des jauges de contraintes, on a suivi, dans un acier pour capacités sous pression, l'évolution des déformations près de l'extrémité d'une fissure de clivage en cours d'arrêt. Les mesures ont été faites en recourant à une disposition de jauges en ligne, située le long due parcours obligé de la fissure et à une distance de 0.65 × l'épaisseur de l'éprouvette au-dessus du plan de fissuration. Une instrumentation de lecture des indications des jauges permettait de déceler les variations de déformations toutes les 2 à 4 micro-secondes. Les résultats ont été collationnés pour une tôle forte suffisamment grande pour que les contributions des ondes élastiques à l'enregistrement des déformations ou à la croissance de la fissure durant cet intervalle de temps soient peu significatives.Dans les limites de résolution propres à cette technique, il est apparu que les jauges de contraintes ne décelaient pas de plastification près de l'extrémité de la fissure de clivage en cours de propagation rapide. Cependant, au moment de l'arrêt, les jauges détectent la formation d'une zone plastique, dont la taille s'accroît sur un intervalle de plusieurs millisecondes.On a interprété cette croissance en termes d'une frontière élasto-plastique qui émane de l'extrémité de la fissure lorsqu'elle s'arrête, et qui atteint ensuite un état d'équilibre, tel que celui prédit par le facteur d'intensité de contraintes appliqué, et par les propriétés plastiques du matériau en conditions statiques.On présente une analyse simple en vue d'estimer la vitesse de cette frontière élasto-plastique à partir de données de déformations, et des résultats relatifs à plusieurs essais sont fournis.

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National Institute of Standards and Techonology

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National Institute of Standards and Techonology     

Plastic zone correction on a Zener–Stroh crack interacting with a circular inclusion in ductile materials

Elastic–plastic stress analysis on a matrix Zener–Stroh crack interacting with a circular inclusion (fibre) in fibre‐reinforced composites has been carried out. The Zener–Stroh crack is initiated near the fibre in the pure matrix. Plastic zone correction is introduced the first time for such a crack–inclusion interaction problem so that the fracture behaviour can be analysed more accurately. To determine the plastic zone sizes, a generalized Irwin model is proposed for the mixed‐mode problem where the Von Mises stress yielding criterion is employed. Different to a Griffith crack, a Zener–Stroh crack propagation always occurs from the sharp tip whose relative position to the inclusion has great effect on the elastic–plastic fracture behaviour of the crack. In our study, the plastic zone size (PZS), crack tip opening displacement (CTOD) and effective stress intensity factor (SIF) are evaluated by solving the formulated singular integral equations. Through the numerical examples, the influence of the inclusion (fibre) shear modulus, crack–inclusion distance and the crack sharp tip position on the fracture behaviour of the crack is discussed. It is found that the shear modulus ratio and the crack–inclusion distance have great effect on the normalized values of PZS and CTOD, but the effects highly depend on the crack sharp tip position.     

The interface crack with a contact zone

An analytical method is applied to the contact zone model of the finite length interface crack problem. The results are valid for small contact zone lengths due to interpenetration of the crack faces. Qualitative results obtained in [1] are confirmed. Also where possible numerical comparisons are made with the work of Comninou et al. [2, 3, 4, 5] and reasonable agreement is obtained.

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Résumé On a appliqué une méthode analytique au modèle de zône de contact dans le cas d'un problème de fissure d'interface de longueur finie. Les résultats sont valables pour des longueurs de la zône de contact faibles, dues à une interpénétration des faces de la fissure. Des résultats qualitatifs obtenus dans la référence [1] sont confirmés. En outre, il a été possible de procéder à des comparaisons numériques avec les travaux de Comninou et al. [Réf. 2 à 5] et un accord raisonnable a été obtenu.

     

Plastic zone near the crack tip in a material with strain anisotropy caused by loading along spatial rectilinear trajectories

The equation of the boundary of the plastic zone near the tip of a mode I crack is deduced for the case of a plate made of a material with strain anisotropy. It is assumed that the anisotropy is caused by hardening in the process of plastic deformation performed prior to the appearance of the crack under loading along arbitrary rectilinear trajectories in the space of the stress tensor. An analysis of this equation demonstrates that the main factors affecting the shape and size of the plastic zone are the level of plastic strains accumulated in the process of preloading, their sign, and the orientation of the crack relative to the axes of anisotropy. Translated from Problemy Prochnosti, No. 6, pp. 21–27, November–December, 1997.     

Plastic zone near the crack tip in a material with strain anisotropy caused by loading along spatial rectilinear trajectories

The equation of the boundary of the plastic zone near the tip of a mode I crack is deduced for the case of a plate made of a material with strain anisotropy. It is assumed that the anisotropy is caused by hardening in the process of plastic deformation performed prior to the appearance of the crack under loading along arbitrary rectilinear trajectories in the space of the stress tensor. An analysis of this equation demonstrates that the main factors affecting the shape and size of the plastic zone are the level of plastic strains accumulated in the process of preloading, their sign, and the orientation of the crack relative to the axes of anisotropy.     

The kinematics of an active zone during fatigue crack layer growth in polystyrene

An investigation of the kinematics of an active zone (or process zone) evolution in polystyrene during fatigue fracture is reported. Experiments were conducted on tension-tension singleedge-notched specimens of 0.25 mm thickness. Craze characterization was carried out on thinned sections of the active zone at six consecutive configurations. Analysis consisted of quantitative comparison of ratios of the inertia moments of the active zone at consecutive configurations. The results indicate that for the particular loading history considered, damage evolution can be approximated by a linear transformation of the space variables. The fracture process can be described by the translation and deformation of the active zone. Consequently, the corresponding energy release rates can be expressed by the J ₁, M and N _ij integrals. The results of this analysis agree with the kinematics proposed by the crack layer model.     

Plastic zone formation and fatigue crack extension during high cyclic bending of steels

In repeated high cyclic bending, with constant load amplitude, the size and the shape of the plastic zone preceding the propagating crack is controlled by local structural conditions near the tip rather than by stress intensity. No significant correlations were found between the experimentally determined sizes of plastic zone and the theoretically predicted values of Liu and Rice. The plastic zone sizes ahead of the propagating crack cannot be simply expressed as proportional to the rate of fatigue crack propagation, though a simple relationship exists between the rate and the stress intensity factor. The relationship given by Paris, dl/dN = QΔKⁿ, describes the rate of crack propagation only in a limited range of relative crack length, x < 0.5. The extent of this range depends on the structure and on the level of applied cyclic stress. Beyond this range, the Paris equation could not be applied and the crack propagation cannot be related to the stress intensity factor.     

Fracture of a finite piezoelectric layer with a penny-shaped crack

This paper studies a penny-shaped crack in a finite thickness piezoelectric material layer. The piezoelectric medium is subjected to a thermal flux on its top and bottom surfaces. Both thermally insulated crack and heated crack are considered. Numerical solution for the finite layer thickness is obtained through the solution of a pair of dual integral equations. The result reduces to the closed form solution when the thickness of the piezoelectric layer becomes infinite. Exact expressions for the stress and electric displacement at the crack border are given as a function of the stress intensity factor, which is determined by the applied thermal flux. This paper is useful for the reliability design of piezoelectric materials in thermal environments.     

Interface crack interacting with a microvoid in the near-tip process zone

This paper deals with the elastic problem of an interface macrocrack interacting with a near-tip microvoid in the process zone. The `pseudo-traction-edge-dislocation method' (abbreviated PTEDM) proposed by Zhao and Chen (1996) is extended to treat the macrocrack-microvoid interaction problem in bimaterial solids. A system of singular integral equations with Hilbert kernel is reduced and solved numerically, which provides a fundamental understanding of the interaction effect. Numerical results are given for six kinds of bimaterial combinations (Hutchinson, Meer and Rice, 1987). The J-integral analysis is also applied to the interaction problem, and some useful conclusions are presented in detail.     

Damage analysis of a crack layer

Damage analysis of a crack layer in polystyrene is carried out by employing optical microscopy and principles of quantitative steteology. The results show that, within the quasistatic phase of crack layer propagation, the average crazing density, along the trailing edge of the active zone, is constant. This is consistent with a self-similarity hypothesis of damage evolution employed by the crack layer theory. The average crazing densities within the active zone and along its trailing edge are found to be practically equal. A layer of constant crazing density, adjacent to the crack planes, accompanies the crack during its quasi-static growth. This suggests that: (1) a certain level of crazing density should be reached, around the crack tip, prior to crack advance; (2) the specific energy, associated with this ‘core’ of damage, could be considered as a Griffith's type energy. The results are in favour of certain hypotheses adopted by the crack layer theory.     

裂尖具线性分布约束应力的运动裂纹模型及其解析解

以恒定速度运动的Griffith裂纹解析解为著名的Yoffe解。静止裂纹的条状屈服模型即Dugdale模型,将其推广到运动裂纹模型时发现,当裂纹运动速度跨越Rayliegh波速时,裂纹张开位移COD趋于(∞,且表现为间断。通过在裂尖引入一个约束应力区及两个速度效应函数,假设约束应力为线性分布,采用复变函数方法,求得动态应力强度因子SIF与裂纹张开位移COD的解析解。新的结果,在Rayleigh波速下裂纹张开位移连续且为有限值。给出裂纹张开位移的一些数值结果,获得了一些有意义的结论。     

Antiplane vibration of an elastic layer with a midplane crack

The oscillations of an elastic layer excited by antiplane tractions of uniform intensity along the faces of a midplane crack are investigated. A singular integral equation is derived and solved numerically by Gauss-Chebyshev integration technique. The stress intensity factor and crack surface response are computed as functions of frequency and layer thickness-to-crack length ratio, and the resonant frequencies of the cracked layer are determined.     

Antiplane shear crack problem in bonded materials with a graded interfacial zone

In this paper the interface crack problem for two elastic half spaces bonded through a nonhomogeneous interfacial zone is considered. It is assumed that the medium is under antiplane shear loading. The problem is solved for two different interfacial zone models that may approximate the actual diffusion bonded materials or homogeneous solids bonded through a functionally gradient material. Extensive results are obtained by varying the stiffness and the interfacial zone thickness to crack length ratios. Also, for various limiting cases the behaviour of the stress intensity factors and the strain energy release rates are studied.     

Plastic energy dissipation due to a penny-shaped crack

International Journal of Fracture - A penny-shaped crack in a material which is ideally elastic-plastic has been envisaged with the assumption that the plastic zone forms a very thin layer...     

Plastic deformation at a stably growing crack tip

The deformations around the tip of a straight, through-the-thickness crack in sheets of cellulose nitrate, celluloid, in tension have been studied, as well before as after the onset of stable crack growth. A tensile testing apparatus and a device for engraving a fine grid directly on the specimen sheet have been designed. The development of the plastic enclave also after the start of stable crack extension has been studied in connexion with investigations of the deformations in gross fields of size up to 30 × 40 mm² performed on a digital computer (line element length 1 or 0.5 mm, virgin crack length 80 mm). Unexpected deformations very close to the crack tip were found in the near field where line elements of length 50 µm were analyzed. A modified Dugdale model where the cohesive stresses vary linearly is examined and is applied on polyvinylehloride, PVC, specimen the thickness of which is strongly reduced near the crack tip. A method for measurement of thickness changes of the order of a few µm by tracing isohypses is described. Finally, to enable comparison with the experimental results, the elastic deformation pattern for a square mesh in the plane strain and plane stress case is obtained by a digital computer.

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Zusammenfassung Es wurden die Verformungen an der Spitze eines geraden, über die gesamte Dicke durchgehenden Risses in Spannungen unterworfenen Tafeln aus Zellulosenitrat (Zelluloïd) untersucht. Dies geschah sowohl vor als auch nach der Einstellung einer gleichmäßigen Rißfortpflanzungsgeschwindigkeit. Hierzu wurde eine Zugversuchsapparatur sowie ein Verfahren zur Gravierung eines Rasters, direkt auf die Probe, entwickelt.Die Ausbildung der plastischen Enklave, auch nach Einstellung einer gleichmäßigen Rißausweitung, wurde in Verbindung mit Studien der Verformung in Grobrastern his 30 × 40 mm mittels Digitalrechnern, untersucht (Länge einer Elementarlinie l oder 0,5 mm; Originallänge des Risses 80 mm). Direkt an der Rißspitze, wo Elementarlinien mit einer Länge von 50 µm untersucht wurden, konme man unerwartete Verformungen feststellen. Es wurde ein abgewandeltes Dugdale-Modell bei dem die Kohäsionsspannungen sich linear verändern untersucht and auf Polyvinylchlorid PVC-Proben, deren Dicke sich an der Rißspitze stark reduziert, angewandt. Es wird ein Verfahren zur Messung von Änderungen der Dicke um wenige µm beschrieben. Um einen Vergleich mit den experimentellen Ergebnissen zu ermöglichen, wird zum Schluß mit Hilfe eines Digitalrechners das elastische Verformungsmodell für Quadratraster, im Falle planer Verformung and eines planen Spannungszustandes berechnet.

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Résumé On a étudié les déformations qui se produisent au voisinage de l'extrémité d'une fissure droite et traversante dans des feuilles de nitrate de cellulose (celluloid) soumises à traction, et ce avant et après établissement d'un régime stable de propagation. A cet effet, on a mis au point un équipement d'essai de traction et une technique permettant de tracer une grille directement sur l'éprouvette. Le développement d'une enclave plastique dès que l'éxtension de la fissure devient stable a été etudie en recherchant les déformations survenant dans des réseaux grossiers, allant jusqu'à 30 × 40 mm², a l'áide d'un calculateur digital (longueur d'une ligne élémentaire: 1 ou 0,5 mm, longueur de fissure initiale: 80 mm). Dans un réseau plus fin, où les lignes élémentaires avaient une longueur de 50 microns, on a analysé, au voisinage immédiat de l'éxtrémité de la fissure, des déformations inattendues.On a examiné l'ápplication d'un modèle du Dugdale modifié, où les contraintes de cohésion varient linéairement, à des éprouvettes de PVC, dont l'épaisseur subit des réductions considérables près de l'extrémité de la fissure. On décrit une méthode de mesure des variations d'épaisseur de l'órdre de quelques microns, qui est basée sur le tracé de courbes de niveaux.Enfin, pour permettre une comparaison avec les résultats expérimentaux, on a établi par un calculateur digital l'állure de la déformation élastique dans un réseau carré pour les cas d'état plan de déformation et d'état plan de tension.