Large strain field near an interface crack tip

In this paper the elastostatic field near the tip of an interface crack between two materials is analyzed with the fully nonlinear theory. By dividing the crack tip field into narrowing sectors and an expanding sector, the asymptotic equations for the crack tip field are derived and solved. The singular characters of stress and strain near the crack tip are revealed. The crack opening shape is discussed for various cases. It is shown that when large deformation is taken into account the oscillatory singularity does not occur.     

疲劳裂纹扩展中裂纹尖端应变变化的详细考察

传统的ASTM2%偏差法和卸载弹性法是以柔度轨迹的线性区段确定开闭口载荷(Pop和Pcl),往往因人工误差导致结果相差很大.本文注重裂纹尖端塑性变形带给柔度变化的物理意义,描述了一个载荷循环下柔度变化与裂纹开闭口以及弹塑性行为的关系.采用自行开发的高精度局部柔度测量法,针对结构钢进行了疲劳试验,记录了裂纹尖端应变与载荷关系的系列迟滞回线,并以微分法定量求出迟滞回线上的特征载荷.根据试验考察结果,分析了文献中几种疲劳裂纹扩展参量的关系.结果表明,ΔKRPG比ΔKcl和ΔKop更适合作为裂纹扩展驱动力参量.     

Higher order asymptotic field at the moving crack tip

The higher order asymptotic field at a smoothly curved dynamic crack tip under mixed loading conditions is investigated by the use of a moving coordinate system and the mathematical framework of the complex potential theory. On the basis of the general representation for stress functions in a moving coordinate system, the recurrence formulae for determining the higher order solutions from lower order ones are derived. The calculation in this paper shows that the higher order asymptotic field can be separated into two parts: the steady state asymptotic field which depends on the crack velocity; and the non-steady state asymptotic field which is determined by the time rate of change of the intensity factor, the crack tip acceleration and rotation speed. The second order solution giving the sress distributions at the moving crack tip are presented. The important result that the experimentally observed crack branching velocities are estimated to be much smaller than Yoffe's prediction might be explained by this second order asymptotic solution.     

The boundary-layer effect on the crack tip field in mechanism-based strain gradient plasticity

The theory of mechanism-based strain gradient (MSG) plasticity involves two material length parameters, namely the intrinsic material length land the mesoscale cell size l , which are on the order of a few microns and 0.1 m, respectively. Prior studies suggest that l has essentially no effect on the macroscopic quantities, but it may affect the local stress distribution. We demonstrate in this paper that there is a boundary layer effect associated with l in MSG plasticity, and the thickness of the boundary layer is on the order of   l ² big/l. By neglecting this boundary layer effect, a stress-dominated asymptotic field around a crack tip in MSG plasticity is obtained. This asymptotic field is valid at a distance to the crack tip between l and l(i.e., from 0.1 m to a few microns). The stress in this asymptotic field has an approximate singularity of r ^–2/3, which is more singular than not only the HRR field in classical plasticity but also the classical elastic Kfield (r ^–1/2). The stress level in this asymptotic field is two to three times higher than the HRR field, which provides an alternative mechanism for cleavage fracture in ductile materials observed in experiments.     

The crack tip strain field of AISI 4340 Part II Experimental measurements

Crack tip strain maps have been measured for AISI 4340 high strength steel. No significant creep was observed. The measured values of CTOD were greater than expected from the HRR model. Crack tip branching was observed in every experiment. The direction of crack branching was in the same direction as a major ridge#x201D; of _yy strain, which in turn was in the same direction as predicted by the HRR model. Furthermore, the measured magnitudes of the _yy strain in this same direction were in general greater than the values predicted by the HRR model. This indicates more plasticity in the crack tip region than expected from the HRR model. This greater plasticity could be related to the larger than expected CTOD values. The following discrepancies between the measured strain fields for AISI 4340 and the HRR predictions are noteworthy: (1) The crack branching. (2) Values of CTOD significantly higher than predicted by HRR. (3) The major ridge of _yy strain an angle of about 60° with the direction of overall propagation of the fatigue precrack, in which the measured magnitudes of the _yy strain were greater than the values predicted by the HRR model. (4) Asymmetric shape of the plastic zone as measured by the _yy strain. (5) Values of shear strain _xy significantly higher than predicted by the HRR model.     

The crack tip strain field of AISI 4340 Part I Measurement technique

This is the first paper in a study on the influence of the environment on the crack tip strain field for AISI 4340. A stressing stage for the environmental scanning electron microscope (ESEM) was constructed which was capable of applying loads up to 60 kN to fracture-mechanics samples. The measurement of the crack tip strain field required preparation (by electron lithography or chemical etching) of a system of reference points spaced at 5 m intervals on the sample surface, loading the sample inside an electron microscope, image processing procedures to measure the displacement at each reference point and calculation of the strain field. Two algorithms to calculate strain were evaluated. Possible sources of errors were calculation errors due to the algorithm, errors inherent in the image processing procedure and errors due to the limited precision of the displacement measurements. Estimation of the contribution of each source of error was performed. The technique allows measurement of the crack tip strain field over an area of 50×40 m with a strain precision better than ±0.02 at distances larger than 5 m from the crack tip.     

The crack tip strain field of AISI 4340 Part III Hydrogen influence

This paper studied the influence of hydrogen and water vapour environments on the plastic behaviour in the vicinity of the crack tip for AISI 4340. Hydrogen and water vapour (at a pressure of 15 Torr) significantly increased the crack tip opening displacement. The crack tip strain distribution in 15 Torr hydrogen was significantly different to that measured in vacuum. In the presence of sufficient hydrogen, the plastic zone was larger, was elongated in the direction of crack propagation and moreover there was significant creep. These observations support the hydrogen enhanced localised plasticity model for hydrogen embrittlement in this steel. The strain distribution in the presence of water vapour also suggests that SCC in AISI 4340 occurs via the hydrogen enhanced localised plasticity mechanism.     

Analysis of crack growth rate based on rotation at the crack tip plastic zone

The crack growth rate of a line crack is obtained from a linear elastic analysis of work required in the formation of a crack tip plastic zone. Equations of crack growth rate are derived from rigid body rotation at the root of Dugdale's plastic zone. The proposed relations are used to study the fracture behaviour of materials in tension tests and the three point bending tests. This revised version was published online in August 2006 with corrections to the Cover Date.     

The crack tip opening angle associated with the growth of a crack in a nonuniform stress field

A general analysis gives the crack tip opening angle associated with the growth of a crack in an infinite solid that is subject to a nonuniform stress distribution. The results allow one to study the effect of nonuniform stresses e.g. residual stresses, on inert environment crack growth and also environmentally assisted crack growth.     

Plastic yield at the tip of an arbitrarily oriented crack under plane strain

The method of discontinuous displacements is used to analyze the plane elastoplastic strain of a body with an arbitrarily oriented rectilinear crack. It is assumed that the areas of plastic deformation at each tip of the crack are located along two narrow strips whose length and orientation are determined in a self-consistent manner. The problem is reduced to a set of three singular integral equations and solved numerically in the cases of a crack under constant pressure and under tensile stress at infinity.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 29, No. 6, pp. 93–96, November–December, 1993.     

Plastic strain distribution at the tip of a fatigue crack. application to fatigue crack closure in the threshold regime

The fatigue crack growth rate behaviour of a Co-33 wt pct Ni alloy was investigated at room temperature down to the threshold regime using CT specimens for two load ratios 0.1 and 0.3. Cyclic equivalent plastic strain distributions along an axis normal to the crack plane were experimentally determined over the whole range of crack growth rates using two techniques microhardness and a quantitative metallographic technique applied to twins, both calibrated on low cycle fatigue specimens. These experimental values were compared with theoretical curves as obtained from the monotonic plane strain finite element analysis of Tracey and adapted to cyclic loading according to the procedure proposed by Rice. A good agreement was found in stage II crack growth in the vicinity of the crack tip but a discrepancy was observed in the low crack growth rate regime, indicative of crack closure. It was possible to determine the effective amplitude of the stress intensity factor which accounts for this discrepancy and an intrinsic crack growth law was obtained which obeys Paris equation and which applies in the whole crack growth rate range independently of the load ratio.     

A comparison between the elastic strain energy and the plastic work at the crack tip

For ductile fractures it becomes progressively imperative to include the plastic components of the quantities defining each particular fracture criterion, in order to derive a relationship, which not only fits the experimental data but also satisfies some appropriate physical principle. In this paper the plastic components of stresses entering in the experessions for the specific plastic work were derived from the plastic singular solution described by the HRR stress field. In order to ascertain the accuracy of such approximate solutions for criteria based on energy balance, a comparison between the components of elastic strain energy density, and its total value, with the specific plastic work (W_p) was undertaken, in order to derive certain conclusions about the importance of the contribution of energy component and the specific plastic work. The analysis was based on the typical case of a transverse internal crack in plate under plane-stress conditions, which is submitted to a mode I deformation. While the dilatational (T_v) and the distortional (T_D) components of elastic SED were calculated for impending plasticity the specific plastic work (W_p) was evaluated by assuming ssy conditions and the theory of HRR field using the plastic singular solution for mode I. It was shown that the contributions of T_v, T_D, T_e (T_e = t_v + T_D) and W_P for different strain hardening exponents were varying with ductile materials, presenting a strong influence of W_p. The conclusion was that for highly strain-hardened materials the contribution of the elastic SED and particularly of T_v is important to the mode of fracture of the plate.     

Fracture at the crack tip in impact loading

Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 4, pp. 75–77, July–August, 1988.     

On distribution of the plastic strain around the crack tip at the onset of ductile fracture

A very simple model is presented to explain the change in the plastic strain from the crack tip. It is based on a relaxation mechanism by dislocations.

---

Zusammenfassung Ein sehr einfaches Modell wird eingebracht zur erklärung der Änderung in der plastischen Verbiegung von der Rißspitze. Es beruht auf einer Relaxationsmechanik durch Versetzungen.

---

Résumé Un modèle très simple est présenté pour expliquer le changement dans le gauchissement plastique dès la pointe fente. C'est conformé à un méchanisme de relaxation par dislocations.

     

Plastic yield on inclined slip planes at the tip of a crack deformed in anti-plane strain

The characteristics of the plastic enclave at the tip of a crack in a medium subjected to stress are investigated by means of a dislocation model. A model in which dislocations can glide along a plane coplanar with the crack has previously been given by Bilby, Cottrell and Swinden⁽¹⁾, and it is the purpose of this work to extend these authors' results to more general situations. We first of all consider the case of a single slip plane at the crack tip which is inclined to the plane of the crack, and develop an approximation technique for solving the equations of equilibrium of the dislocations. This yields the size of the plastic zone, the density of dislocations and the plastic displacement at the crack tip as explicit functions of the crack-length, applied stress, angle between crack and slip-plane and yield stress and elastic modulus of the medium. These functions become particularly simple for either low or high stresses. A more difficult problem is that in which slip can occur on two different slip-planes at the crack-tip. The same approximation method can be used, but the results are no longer expressible in terms of elementary functions. A computer program has been used to calculate numerical solutions in some cases and in one case to evaluate higher approximations to the solution.

---

Zusammenfassung Untersucht werden die wesentlichen Merkmale der plastischen Zone am Ende eines Risses in einem, einer Scherungskraft ausgesetzten Stoff. Dies geschieht mittels eines Versetzungsmodells. Es wird eine frühere Methode von Bilby, Cottrell and Swinden⁽¹⁾ verallgemeinert für die Fälle einer and zweier Gleitebenen am Rissende, die nicht mit dem Riss in einer Ebene liegen. Eine Approximations-Technik liefert die Grösse der plastischen Zone ausgedrükt durch Risslänge, angewandte Kraft and Gleitflächenwinkel. Im zweiten Fall ist numerische Behandlung erforderlich.

---

Résumé Les caractéristiques d'une enclave plastique á l'extrémité d'une félure dans un milieu soumis à l'effort sont étudiées au moyen d'un modèle de dislocation. Une méthode antérieure due à Bilby, Cottrell et Swinden est étendue aux cas d'un et de deux jeux de plans de glissement a l'extrémité de la félure qui ne sont pas dans le même plan que la félure. Une technique d'approximation fournit la dimension de la zone plastique selon la longueur de la félure, la force appliquée et les angles des plans de glissement. Dans le deuxième cas, une évaluation numérique est nécessaire.

     

Displacements and strains at the crack tip

Displacement and strains in the plastically yielded region generated ahead of a crack were measured with replica technique. The shape and length of the yielded zone were detected on mild steel plates of various thicknesses.The results of plastic zone length and displacement measurements were compared with theoretical solutions of both DBCS model and infinite series of identical coplanar cracks. The measured values of these quantities are, below the general yield, lower than theoretically predicted, and decrease with increasing plate thickness. From this there follows that the plastic zone length and crack tip displacement depends on the degree of stress triaxiality and the theoretical solutions for plane stress or anti-plane strain cannot be used for situations approaching plane strain.The comparison of these results with fracture stress vs. temperature curves suggests that critical value of crack tip displacement as fracture criterion can be used only for bodies fractured after general yield.

---

Zusammenfassung Verschiebungen und Verformungen in derv Fliessgebiet an einer Risswurzel wurden unter Anwendung des Abdruck-Verfahrens gemessen. Die Form und Lange dieses Fliessgebietes wurden an Weichstablplatten verschiedener Dicke ermittelt.Die Messwerte der Langen des Fliessgebietes und der Verschiebungen wurden mit den rechnerisch ermittelten Losungen von DBCS Modell und unendlichen Reihen identischer Komplanarer Risse verglichen. Die gemessenen Werte dieser Grossen sind, im Bereiche unter dem Fliessen im gesamten Querschnitt (general yield), kleiner als die theoretisch vorgesehenen Werte und nehmen mit zunehmender Plattendicke ab. Daraus kann geschlossen werden, dass die Länge des Fliessgebiets sowie auch die Verschiebung an der Risswurzel, von der Spannungs-Triaxialität abhängon und dass die theoretischen Lösungen für ebene Spannung oder Längsschub night in den Fällen gebraucht werden können, die dem Zustand einer ebenen Verzerrung nahe liegen.Ein Vergleich von obenerwähnten Ergebnissen mit den Kurven der Abhängigkeit der Rissfestigkeit von der Temperatur zeigt, dass der kritische Wert der Verschiebung an der Risswurzel nur bei den Körpern als gas Kriterium eines Bruches betrachtet werden kann, wo der Bruch nach Fliessen im gedamten Querschitt erfolgt.

---

Résumé Déplacements et déformations plastiques dans la zone au devant de la racine d'une fissure ont été mesurés en se servant de la méthode de réplique. La forme et la longueur de la zone soumise à la déformation plastiques ont été déterminées sur plaques en acier doux d'épaissaures différentes.Les résultats des mesures de la longueur de la zone plastique, de méme que des déplacements, ont été compares aux résultats de solutions théoriques de modèle DBCS d'un côte et des series infinies des fissures identiques disposées au même plane de l'autre côté. Les valeurs measures de ces quantities sont, dans la gamme sous la déformation plastiques de la section totale (general yield), inférieurs au prevues par la théorie et diminuent au fur et à mesure que l'épaisser des plaques s' augmente. Il s'en suit que la longueur de la zone plastique, de méme que le déplacement au devant de la racine d'une fissure, dépendent du degré de trixialité de la contrainte appliquée et que les solutions théorique concues pour les cas d'une contrain et plane ou déformation anti-plane ne sauraient etre appliquées dans les situations qui s'approchent de l'état de déformation plane.En comparant les résultats ei—dessus avec les courbes de dépendence de la containte de rupture de témperature, on voit que la valeur critique du déplacement au devant de la racine d'une fissure ne saurait être consid'ereé comme critère de rupture qu'en cas ou la rupture survient après une déformation plastique de la section totale.

     

The distribution of strain within crack tip plastic zones

Strains ahead of cracks both growing due to cyclic loading and loaded only in tension, as experimentally determined using the stereoimaging technique, have been used to derive equations for the strain distribution. Both logarithmic and power function relations were fitted to the strains and the logarithmic function was found to be the best fit to most data. Materials examined were 7075 and 7091 aluminum alloys, Ti-6Al-4V titanium alloy and austenitic 304 stainless steel.