Effects of short cracks produced at blunted pre-crack tip on stress and strain distributions

The present work investigates problems: (1) How are the plastic strain and the stress (triaxiality) re-distributed after a short crack initiated, extended and blunted at the pre-crack tip? (2) How do the above changes put a crucial effect on the triggering of the cleavage fracture? Based on the previous observations of configuration changes and fracture surfaces of pre-crack tips, Finite element method (FEM) simulations of a short crack initiated, extended and blunted at a pre-crack tip and calculations of distributions of stress, strain and triaxiality are carried out for 3PB pre-cracked HSLA steel specimens tested at -130^°C. The results reveal that: as long as the fatigue pre-crack is only blunted, in its vicinity a region where the accumulated strain is sufficient to nucleate a crack, and a region where the stress (triaxiality) is sufficient to propagate a crack nucleus are separated by a distance. The nucleated crack cannot be propagated and the cleavage fracture cannot be triggered. While a short crack produced at the fully blunted fatigue pre-crack, the strain retains, the stress (triaxiality) is rebuilt. An initiated and significantly extended and then blunted short crack makes a tip configuration, which on one hand is much sharper than that of the fully blunted original pre-crack tip, on other hand is wide enough to spread its effects into the high stress covered region. This sharpened crack tip configuration re-builds a ‘sharper’ distribution of stress (triaxiality) and makes two regions metioned above closer. Finally the two regions overlap each other and a cleavage crack can be initiated and propagated at a distance ahead of the blunted fatigue pre-crack.     

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.     

The elastic stress fields of elliptic and tapered cracks with predefined shapes

The shape of a tapered crack is more alike cracks in brittle materials than an elliptical crack. The deformation and stress fields for a tapered crack are therefore estimated for hydrostatic pressure and tensional stress by applying the method of complex potentials. The stress fields for the tapered and elliptical cracks are quite similar, which suggests that the elliptical crack can be used as a model for the stress fields for cracks in general. However, the tapered crack has a larger tensional stress at the crack tip, which show that fracture propagation occur at lower applied stresses than for the elliptical crack. A tapered shape of fluid filled fractures can account for their discontinuous propagation. The discontinuous fracture propagation is observed in a large scale by volcanic eruptions where the fracture propagation generates seismic activity.     

Plane strain asymptotic fields for cracks terminating at the interface between elastic and pressure-sensitive dilatant materials

The plane strain asymptotic fields for cracks terminating at the interface between elastic and pressure-sensitive dilatant material are investigated in this paper. Applying the stress-strain relation for the pressure-sensitive dilatant material, we have obtained an exact asymptotic solution for the plane strain tip fields for two types of cracks, one of which lies in the pressure-sensitive dilatant material and the other in the elastic material and their tips touch both the bimaterial interface. In cases, numerical results show that the singularity and the angular variations of the fields obtained depend on the material hardening exponent n, the pressure sensitivity parameter μ and geometrical parameter λ. This revised version was published online in August 2006 with corrections to the Cover Date.     

The elastic interaction between solute atoms and stress field at the tip of cracks and notches

As for a mechanical interaction between a general stress singularity and an-environmental factor (e.g. hydrogen) which has influence on delayed failure characteristics of materials, it has not hitherto been analysed, although it is an important problem. In the present article an analytical basis is given for the problem of elastic interaction between solute atom as an environmental factor accelerating delayed failure and stress field around a crack and further a general V-shaped notch.The result of analysis shows that the number of solute atoms which accumulate to crack tip due to elastic interaction is proportional to a parameter. $\text{ρ}{}_0\text{(}\text{DKt}\text{kT}\text{)}{}^{\mathrm{<mtext>4</mtext><mtext>5</mtext>}}$ where K is stress intensity factor,ρ₀ initial uniform density of the solute atom, D diffusion constant, t time, k Boltzman's constant and T absolute temperature. It is further shown that the abovementioned parameter is generalized to $\text{ρ}{}_0\text{(}\text{Dkt}\text{kT}\text{)}{}^{\mathrm{<mtext>2</mtext><mtext>(2+q)</mtext>}}$ for the number of what accumulate to the apex of a general V-shaped notch due to elastic interaction where k is a factor representing a stress singularity of the notch apex and q is a constant related to apex angle a.The above analysis indicates that it is possible to describe development of delayed failure in terms of a parametre of fracture mechanics K, when viewed from a standpoint inclusive of diffusion and mechanical processes, since it is shown that the rate of concentration to crack tip of solute atoms as an environmental factor is uniquely determined by a factor $\text{ρ}{}_0\text{(}\text{DK}\text{kT}\text{)}{}^{\mathrm{<mtext>4</mtext><mtext>5</mtext>}}$.     

Use of modified standard 20-node isoparametric brick elements for representing stress/strain fields at a crack tip for elastic and perfectly plastic material

The analysis of defects in engineering structures and components has to take into account the singular strain field at the crack tip. The problems encountered in such analyses have unique geometries, have some non-linear elastic plastic behaviour and are three-dimensional in nature. Their solution calls for the use of the finite element method. Two-dimensional fracture mechanics analysis methods have been developed and proved by other researchers to show that 8-noded collapsed finite elements have the required singular strain fields for both the elastic and perfectly plastic material conditions. This paper proves the conditions under which three-dimensional collapsed elements represent the stress/strain fields at a crack tip required for elastic and perfectly plastic material, including crack tip blunting in the latter case. The collapsed elements presented can be used with confidence and give large savings in computing time, which is an essential point in three-dimensional finite element calculations.

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Résumé L'analyse des défauts dans les constructions et dans les composants doit prendre en considération le champ singulier de déformation à l'extrémité d'une fissure. Les problèmes rencontrés dans une telle analyse présentent des géométries uniques, un comportement élasto-plastique sensiblement non linéaire et sont tri-dimensionnelles par nature. Leur solution fait appel à l'utilisation de la méthode par éléments finis. Des méthodes d'analyse en mécanique de rupture bi-dimensionnelles ont été développées, et d'autres chercheurs ont établi qu'elles montrent que les éléments finis à 8 noeuds présentent les champs de déformation singuliers, requis à la fois pour les conditions de matériaux élastiques et parfaitement plastiques. La présente étude établit les conditions sous lesquelles des éléments tri-dimensionnels représentent les champs contrainte/déformation à l'extrémité d'une fissure, requis dans le cas de matériaux élastiques et parfaitement plastiques, y compris l'arrondissement de l'extrémité de la fissure pour ce dernier cas. Les éléments présentés peuvent être utilisés avec confiance et procurent de grandes économies de temps dans les calculs, ce qui est essentiel dans le cas des éléments finis à 3 dimensions.

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List of symbols x, y, z coordinates in the real space - g, h, s coordinates in the normalized cubic space - , , coordinates in modified normalized cubic space - u, v, w vectors of displacements inx, y andz direction, respectively - vectors of coordinatesx, y andz - vectors of displacementsu, v andw - N vector of shape functions - M vector of reduced shape functions - vector of stress components - vector of strain components - D matrix containing material properties - J determinant of the Jacobian matrix - J Jacobian matrix - r distance from the crack tip     

Crack-tip fields for matrix cracks between dissimilar elastic and creeping materials

The stress fields near the tip of a matrix crack terminating at and perpendicular to a planar interface under symmetric in-plane loading in plane strain are investigated. The bimaterial interface is formed by a linearly elastic material and an elastic power-law creeping material in which the crack is located. Using generalized expansions at the crack tip in each region and matching the stresses and displacements across the interface in an asymptotic sense, a series asymptotic solution is constructed for the stresses and strain rates near the crack tip. It is found that the stress singularities, to the leading order, are the same in each material; the stress exponent is real. The oscillatory higher-order terms exist in both regions and stress higher-order term with the order of O(r°) appears in the elastic material. The stress exponents and the angular distributions for singular terms and higher order terms are obtained for different creep exponents and material properties in each region. A full agreement between asymptotic solutions and the full-field finite element results for a set of test examples with different times has been obtained.     

Lower bounds on the stress and strain fields inside random two-phase elastic media

Summary The heterogeneous media under consideration are isotropic composites consisting of two well-ordered elastic isotropic phases and subjected to uniform macroscopic loading. By extending a method due to Lipton [2], lower bounds on the stress and strain fields inside each phase are explicitly established in terms of the phase volume fractions and properties. These bounds on the second moments turn out to be optimal, since they are achieved by the relevant stress and strain fields inside the finite-rank laminates which, constructed by Francfort and Murat [6], attain the Hashin-Shtrikman lower and upper bounds on the elastic bulk and shear moduli.     

Evaluation of crack tip stress and strain fields under nonproportional loading in a viscoelastic material

The crack tip strain and stress fields in a viscoelastic material under nonproportional loading conditions are evaluated. In order to evaluate the strain field, the crack tip displacement field is measured by using the digital image correlation (DIC) technique. This displacement field is then approximated by using the theoretically obtained crack tip displacement field in viscoelastic materials. The result shows that the approximation method can smoothly reconstruct the experimentally obtained displacement field. From the approximated displacement field, the crack tip strain field can be precisely obtained by using the differential form of the theoretical displacement. On the other hand, the crack tip stress field is analyzed by using the stress function. This suggests that the strain and stress fields can be independently evaluated. In addition, different time dependencies between stress and strain fields near the crack tip are observed. Based on this experiment, we can discuss the several criteria for the crack propagation directions in viscoelastic materials.     

Plane strain stress intensity factors for branched cracks

A method of calculating stress intensity factors for branched and bent cracks embedded in an infinite body has been developed. The branches are always assumed to be sharp cracks and are modelled by dislocation distributions. The original crack may be either sharp or of elliptical cross-section with finite root radius. Hence, the method which has a precision ±2%, is also applicable to the study of crack branches emanating from elliptical holes and, approximately, also from notches. The following detailed calculations have been made assuming mode I loading: branched sharp crack with branches of equal and different length, bent sharp crack, and one and two crack branches emanating from the crack with a finite root radius. Bending of a sharp crack under mixed mode loading has also been studied. The criteria of maximum tensile stress and maximum energy release rate used in the study of direction of crack propagation are discussed.     

Crack tip stress intensity factor of the double slip plane crack model: Short cracks and short short-cracks

The crack tip stress intensity factorK ₁ for a short crack is determined using the double slip plane (DSP) crack model. It is shown thatK ₁ for a stationary crack is larger than the nominal stress intensity factorK. This result differs from the case of the stationary DSP long crack for whichK ₁ =K. The physical cause whyK ₁ >K is the fall off with distance of the dislocation shielding/antishielding factor I. at a rate faster than an inverse square root dependence when the distance from a dislocation to a crack tip is of the order of or larger than the crack half widtha. The value ofL for a dislocation situated at an arbitrary position about a crack is derived in this paper. (The Rice-Thomson expressions forL are valid only if a dislocation is very close to a crack tip.) The short short-crack is also analysed using the DSP crack model. (A short short-crack crack is defined to be a short crack whose plastic zone behind the crack tip extends to the center of the crack.) The value of K₁ for the short short-crack is a constant and is larger than K. Finally, it is shown that if the crack length is smaller than a critical value that is inversely proportional to the yield stress and is proportional to the critical stress intensity factor K_cb of a Griffith crack that K₁ must be smaller than K,t, regardless of how closely the applied stress approaches the yield stress. These results imply that fatigue crack growth of short cracks in the DSP crack model occurs at a faster rate than for long cracks when the conventional cyclic stress intensity factor is above the threshold value and that short cracks can grow under cyclic stress intensity factors smaller than the threshold value.

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Résumé On détermine le facteur d'intensité d'entailleK ₁, à l'extrémité d'une fissure courte en recourant au modèle de fissure à double plan de glissement DSP). On montre que, pour une fissure stationnaire,K ₁ est plus grand que le facteur d'intensité de contrainte nominalK. Ce résultat se distingue du cas où l'on applique le modèle DSP à une fissure longue, qui conduit àK ₁ =K. La raison physique pour laquelleK ₁ >K réside dans le fait que lorsque la distance qui sépare une dislocation d'une fissure est égale ou supérieure à la demi largeur de la fissure, le facteurL de bloquage/débloquage des dislocations s'estompe rapidement en fonction de la distance.On établit dans l'étude la valeur deL correspondant à une dislocation sise dans une position arbitraire par rapport à une fissure (à noter que les expressions de Rice-Thompson pour L ne sont applicables que si la dislocation est très proche de l'extrémité de la fissure). On étudie également é l'aide du modèle DSP le cas de la fissure courte-courte, que l'on définit comme celle dont la zone plastique derrière son extrémité s'étend jusqu'à son centré. La valeur deK _t pour une fissure courte-courte est une constante et est supérieure à K. Enfin, on montre que si la longueur de la fissure ne dépasse pas une valeur critique, inversement proportionnelle à la contrainte limite d'écoulement et proportionnelle au facteur d'intensité de contraintesK _cb d'une fissure de Griffith, la valeur deK _t doit être inférieure àK _cb, quelque proche de la contrainte limite d'écoulement que soit la contrainte appliquée.Ces résultats impliquent que la vitesse de propagation d'une fissure courte par fatique suivant le modèle DSP est supérieure à celle relative à une fissure longue, lorsque le facteur conventionnel de concentration de contraintes cycliques dépasse une valeur de seuil, et que des fissures courtes peuvent s'étendre sous un facteur d'intensité de contraintes cycliques plus petit que cette valeur de seuil.

     

Local stress and strain fields near a spherical elastic inclusion in a plastically deforming matrix

At a micro-sclae, fracture often starts in the vicinity of inclusions in a deforming matrix where the local stress and strain conditions may lead to either failure of the inclusion/matrix interface or of the particle itself. Analytic solutions are available for the local stress and strain fields near an elastic inclusion in an elastically deforming matrix but for plastic deformation it is necessary to resort to numerical analyses. Here a numerical solution is presented for a spherical elastic inclusion in an elastic/plastic matrix, concentrating largely on the particle/matrix interface which is of relevance to ductile fracture. Solutions are also presented for rigid and elastic inclusions in hardening and non-hardening matrices.

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Résumé A une échelle microscopique, une rupture démarre souvent au voisinage d'inclusions dans une matrice déformée où les conditions de contrainte et de déformation locales peuvent conduire soit à une rupture de l'interface inclusion/matrice ou de la particule elle-même. Des solutions analytiques sont disponibles pour déterminer les champs locaux de contrainte et de déformation au voisinage d'une inclusion élastique dans une matrice élastiquement déformée, mais pour des déformations plastiques, il est nécessaire de recourir à des analyses numériques. On présente ici une solution numérique applicable à une inclusion élastique sphérique dans une matrice élasto-plastique, en se concentrant largement sur l'interface particule/matrice qui est en cause dans la rupture ductile. Des solutions sont également présentées pour des inclusions rigides et élastiques situées dans des matrices en matériau durcissable ou non durcissable.

     

Creep crack growth modeling and near tip stress fields

The problem of near tip stress fields in a cracked body subjected to Mode I loading at elevated temperatures is studied. Specifically, the superalloy, IN 718, is examined in the standard compact tension specimen geometry. The simulation is at 650°C. The specimen is assumed to be under dead load conditions. For a stationary crack, the near tip stress fields are calculated and compared with the asymptotic solutions available in the literature. While the results assuming small strains agree very well with the asymptotic solutions, the large strain analysis does not. The results indicate that both the amplitude and the asymptotic exponent are dependent on the applied load level which is in disagreement with the asymptotic predictions. In addition, the zone effected by creep deformation is larger when large strains are considered. An algorithm is developed and tested for the modeling of stable crack growth. Both convergence and stability are investigated. Explicit time integration is used for crack growth studies as it is demonstrated to be computationally more efficient. The algorithm is employed to study the near tip stress fields for a growing crack. The near tip stress fields for a growing crack (with constant velocity) are generated using the developed algorithm. The results demonstrate that the asymptotic behavior of the stress field is load dependent. Comparison is made with the limited analyses available. Recommendations for future research are discussed.     

A unified approach to the evaluation of linear elastic stress fields in the neighborhood of cracks and notches

The problem of evaluating linear elastic stress fields in the neighborhood of cracks and notches is considered. An analytical solution valid for cracked and notched components is given in general terms, according to Muskhelishvili's method based on complex stress functions. The solution is particularly useful for V-shape notches in wide and finite plates under uniform tensile loading. It will be demonstrated that some remarkable solutions of fracture mechanics and notch analysis already reported in the literature can be considered special cases of this general solution, as soon as appropriate values of the free parameters are adopted.     

Finite elements for determination of crack tip elastic stress intensity factors

A new type of finite element is introduced which embodies the inverse square root singularity present near a crack in an elastic medium. Using this element near the tip in two typical cracked configurations, stress intensity factors within 5 per cent of accepted values were obtained with meshes having as few as 250° of freedom.     

Perturbation solution for near-tip fields of cracks growing in elastic perfectly-plastic compressible materials

With = 1/2 – ( - Poisson's ratio) as a small parameter, perturbation expansion is made, based upon the generally accepted solution for near-tip fields of cracks growing in elastic perfectly-plastic incompressible materials under plane strain. Asymptotic solutions up to the order ² are obtained for near-tip fields of cracks growing quasi-statically and steadily in elastic perfectly-plastic compressible materials. The near-tip field has a 5-sector structure. As 1/2 the 5-sector solution degenerates to the 4-sector solution for the case of incompressible materials. The perturbation expansion provides deeper insight into the process of degeneration of the 5-sector solution to the 4-sector one, and also gives approximate analytical solutions for the case of compressible materials.

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Résumé On réalise l'expansion d'une perturbation en faisant légèrement varier le paramètre = 1/2 – ( = Module de Poisson) et en se basant sur la solution g'enéralement admise pour des champs au voisinage de l'extrémité de fissures en croissance dans des matériaux incompressibles parfaitement élastiques-plastiques soumis á déformation plane.On obtient des solutions asymptotiques jusqu'à l'ordre ² pour des champs au voisinage de l'extrémité de fissures qui croissent de manière quasi-statique et stable dans des matériaux compressibles élastiques-parfaitement plastiques. Ce type de champ a une structure comportant cinq secteurs. Lorsque tend vers 1/2, la solution à cinq secteurs dégénère en la solution à quatre secteurs relative au cas des matériaux imcompressibles.L'expansion de la perturbation procure une vision plus profonde de ce processus de dégénérescence, et donne des solutions analytiques approchèes pour le cas des matériaux compressibles.