Requirements for a one parameter characterization of crack tip fields by the HRR singularity

Detailed crack tip stress and strain fields are generated for the edge crack bar subjected to bending and the center cracked panel and single cracked panel subjected to tensile loads. These fields are compared with the singular fields, due to Hutchinson and Rice and Rosengren, at the same level of applied J. The comparisons show that the size R of the region at the crack tip dominated by the HRR singularity is substantially larger in the bend specimen than in the center-cracked panel for contained and large scale plasticity. Hardening properties also have a strong influence on R. The implications of these results on the minimum size requirement essential to a one parameter fracture criterion based on the J-integral or crack tip opening displacement are discussed.

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Résumé Lorsqu'un barreau fissuré sur ses bords est soumis à flexion, ou lorsqu'un panneau fissuré en son centre ou présentant une fissure simple est soumis à des contraintes de traction, des champs complexes de contrainte et de déformation aux extrémités des fissures considérées apparaissent. Ces champs sont comparés aux champs singuliers étudiés par Hutchinson et Rice et Rosengren pour le même niveau d'un J déterminé. Les comparaisons montrent que la dimension R de la région à l'extrémité d'une fissure sur laquelle agit de manière déterminante une singularité HRR se trouve être substantiellement plus grande dans une éprouvette de flexion que dans un panneau présentant une fissure centrale, et ce pour une plasticité contenue ou une plasticité à large échelle. Les propriétés de durcissement ont également une influence déterminante sur R. On discute les implications de ces résultats sur l'exigence de dimension minimum qui est essentielle dans le cas du critère de rupture à un paramètre basé sur une intégrale J ou dans le cas du COD.

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Appeared in General Electric Technical Information Series publication, October 1978.     

HRR fields for damaged materials

The paper presents an investigation of the interaction between a macroscopic crack and distributed damage in an elastic-plastic material based on the HRR field model for virgin materials. This is achieved by describing the mechanical effects of the distributed micro-cracks in terms of the damage variable D on the HRR fields. Damage evolution equation and the constitutive equations coupled with damage are formulated and the resulting boundary value problems are solved numerically. Material constants , n and m ₀are varied to examine their effects on the resulting stress distributions. It is found that the HRR fields for damaged and virgin materials are surprisingly similar although the severity of damage equivalent stress is of several orders of magnitude higher than the conventional plastic equivalent stress without damage consideration. Furthermore, it is shown theoretically and justified numerically that the J-integral loses its path independency for damaged materials, causing the amplitude of the singularity K _D to remain an unknown variable in the asymptotic analysis.     

Plastic near-tip fields for branched cracks

A procedure is outlined whereby the plastic near-tip stress and deformation fields for branched cracks are determined under plane strain and small scale yielding conditions. This method utilizes the known elastic stress intensity factor solutions and the universal mixed-mode plastic near-tip fields to determine the stress and deformation conditions at the tip of a kinked or forked crack. The plastic near-tip fields are characterized by an amplitude and a mixity parameter. We examine the influence of crack tip plasticity and material strain hardening characteristics on the local stress and strain states. Possible beneficial effects of crack branching and crack tip plasticity on fracture toughness and crack growth resistance are discussed in the light of these results.

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Résumé On met l'accent sur une procédure par laquelle on détermine les contraintes plastiques au voisinage d'extrémités de fissures en branches et les champs de déformation correspondants, sous un état plan de déformation et sous des conditions de plastification à petite échelle.La méthode fait appel à des solutions connues pour le facteur d'intensité de contrainte, ainsi qu'aux champs plastiques de mode mixte au voisinage de l'extrémité d'une fissure pour déterminer les conditions de contrainte et de déformation à la pointe d'une fissure en fragmentation ou d'une fissure en fourche. Les champs plastiques au voisinage de l'extrémité sont caractérisés par un paramètre de mixité et par une amplitude.On examine l'influence de la plasticité à la pointe de la fissure et les caractéristiques d'écrouissage du matériau sous les état locaux de contrainte de dilatation.On discute de la possibilité d'effets bénéfiques d'une fissure efflorescente et de la plasticité à l'extrémité d'une fissure sur la ténacité à la rupture et sur la résistance à la croissance d'une fissure, à la lumière de ces résultats.

     

Elastodynamic near-tip fields for a rapidly propagating interface crack

The elastodynamic stress field near a crack tip rapidly propagating along the interface between two dissimilar isotropic elastic solids is investigated. Both anti-plane and in-plane motions are considered. The anti-plane displacements and the in-plane displacement potentials are sought in the separated forms $\text{r}{}^{\mathrm{q}}\text{F(θ)}$, $\text{r}$ and θ being polar coordinates centered at the moving tip. The mathematical statement of the problem reduces to a second-order linear ordinary differential equation in θ, which can be solved analytically. Formulation of the boundary and interface conditions leads to an eigenvalue problem for the singularity exponent $\text{q}$. For the in-plane problem, root $\text{q}$ is found to be complex. Thus, the stresses exhibit violent oscillations within a small region around the crack tip, and the solutions have physical significance only outside this region. The angular stress distributions are plotted for various crack speeds, and it is found that at a high enough speeds the direction θ of maximum stress moves out of the interface. This result indicates that a running interface crack may move into one of the adjoining materials.     

Surface effects on the near-tip stress fields of a mode-II crack

On the physical nature, most crack tips are not ideally sharp but have a small curvature radius. Both surface energy and crack-root curvature affect the stress and displacement fields in the vicinity of the crack tip. In the present paper, a numerical method, which incorporates the effect of surface elasticity into the finite element method, is employed to study the surface effects on the mode-II crack tip fields. It is found that when the curvature radius of the crack root decreases to micro-/nanometers, surface elasticity has a significant influence on the stresses near the crack tip. For a mode-II crack, surface effects alter both the magnitude and position of the maximum stresses, as is different from a mode-I crack, in which case only the stress magnitude is influence by surface stresses.     

Persistence of the HRR fields for general yielding in mode II

The finite element program BERSAFE is used to analyse an edge-cracked square plate under mode II loading in the elastic-plastic regime. Plane strain constraint and power law hardening withn = 3 are assumed. Crack tip stress and strain fields are investigated and shown to equal the HRR fields at all loads from small scale yielding through into general yielding.

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Résumé Le programme d'éléments finis BERSAFE est utilisé pour l'analyse d'une plaque carrée fissurée à ses bords et soumise en régime élastoplastique à une sollicitation de mode II. On suppose un état plan de déformation et un écrouissage régi par une loi de puissancen = 3. On étudie les champs de contraintes et de déformations à l'éxtrémité des fissures et on montre que ces champs sont équivalents à ceux décrits par Hutchinson, Rice et Rosengren (HRR) pour toute la gamme de mise en charges allant de la plastification locale de faible envergure à la plastification généralisée.

     

The T-criterion for ductile fractures in HRR plastic singular fields

When a material exhibits large amounts of plastic deformation before failure, with a continuous evolution of the elastic-plastic boundaries around the crack tip the assumptions made by all criteria based on linear fracture mechanics (LEFM) are no longer valid.The necessary modifications of the T-criterion, for adapting it to elastic-plastic solutions, are presented here together with an illustrative example, comprising the case of the definition of the T-criterion by means of the singular plastic solutions of Hutchinson, Rice and Rosengren, henceforth denoted HRR, together with its mixed-mode extension studied by Shih [1–4]. It was shown that the T-criterion is still valid for elastic-plastic cases provided the components of energy are appropriately defined

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Résumé Lorsqu'un matériau fait état de grandes déformations plastiques avant rupture, accompagnées d'une évolution continue des frontières entre élasticité et plasticité autour de l'extrémité de la fissure, les hypothèses associées à tous les critères basés sur la théorie de la mécanique de la rupture linéaire et élastique ne sont plus guère valables.Les modifications nécessaires à apporter au critère T pour l'adapter aux solutions élasto-plastiques sont présentées et sont illustrées par un exemple. Ces modifications portent sur la définition du critère T à l'aide des solutions plastiques singulières de Hutchinson, Rice et Rosengren, dénommées HRR, et sur leur extension pour un mode mixte, telle qu'étudiée par Shih.On montre que le critère T est encore valable pour les cas d'élasto-plasticité pour autant que les composantes de l'énergie soient bien définies

     

The strain-hardening effect in HRR plane fields according to T-criterion

The strain-hardening effect on fracture is investigated with the aid of the T-criterion using HRR stress fields [1–3] around a crack tip in a power hardening material. Using the appropriate components of strain energy density for the elastic-plastic as well as a nearly elastic expression of the T-criterion, we find the fracture angles, as well as fracture stresses in materials possessing an elastic extended up to a perfectly plastic behavior, by considering plane mixed-mode deformation at the crack tip.Significant influence of the strain hardening coefficient, n on the fracture stress, as well as the hardening parameter mainly appeared in plane strain conditions. This phenomenon was observed almost independently of the solution applied, which provides a nearly elastic or an elastic-plastic expression of the T-criterion describing the fracture conditions.     

Two-parameter characterization of the near-tip stress fields for a bi-material elastic-plastic interface crack

A particular case of interface cracks is considered. The materials at each side of the interface are assumed to have different yield strength and plastic strain hardening exponent, while elastic properties are identical. The problem is considered to be a relevant idealization of a crack at the fusion line in a weldment. A systematic investigation of the mismatch effect in this bi-material plane strain mode I dominating interface crack has been performed by finite strain finite element analyses. Results for loading causing small scale yielding at the crack tip are described. It is concluded that the near-tip stress field in the forward sector can be separated, at least approximately, into two parts. The first part is characterized by the homogeneous small scale yielding field controlled by J for one of the interface materials, the reference material. The second part which influences the absolute value of stresses at the crack tip and measures the deviation of the fields from the first part can be characterized by a mismatch constraint parameter M. Results have indicated that the second part is a very weak function of distance from the crack tip in the forward sector, and the angular distribution of the second part is only a function of the plastic hardening property of the reference material.     

Determination of the near-tip stress fields for the scattering of P-waves by a crack

Summary A conservation law implied by the field equations of linear elastodynamics is derived, and a procedure based on this conservation law is given for the direct determination of the near-tip stress fields arising from the scattering of normally incident P-waves by a crack in a homogeneous, isotropic elastic medium.     

Time-domain boundary element analysis of dynamic near-tip fields for impact-loaded collinear cracks

A time-domain boundary integral equation method has been developed to calculate elastodynamic fields generated by the incidence of stress (or displacement) pulses on single cracks and systems of two collinear cracks. The system of boundary integral equations has been cast in a form which is amenable to solution by the boundary element method in conjunction with a time-stepping technique. Particular attention has been devoted to dynamic overshoots of the stress intensity factors. Elastodynamic stress intensity factors for pulse incidence on a single crack have been computed as function of time, and they have been compared with results of other authors. For collinear macrocrack-microcrack configurations the stress intensity factors at both tips of the macrocrack have been computed as functions of time for various values of the crack spacing and the relative size of the microcrack.     

Elastodynamic near-tip fields for a crack propagating along the interface of two orthotropic solids

The elastodynamic stress field near a crack tip rapidly propagating along the interface between two dissimilar orthotropic elastic solids is solved numerically, for in-plane motion. The cartesian displacements are sought in the separated forms, $\text{r}{}^{\mathrm{p}}\text{U(θ)}$ and $\text{r}{}^{\mathrm{p}}\text{V(θ)}$, $\text{r}$ and θ being polar coordinates centered at the moving tip. This reduces the mathematical statement of the problem to two complex second-order linear ordinary differential equations for complex functions $\text{U(θ)}$ and $\text{V(θ)}$. By means of the finite difference method, a matrix eigenvalue problem of the type $\text{ΣA}{}_{\mathrm{ij}}\text{(p)X}{}_{\mathrm{j}}\text{= 0}$, is obtained where $\text{A}{}_{\mathrm{ij}}\text{(p)}$ are polynomials of the complex variable $\text{p}$ and $\text{X}{}_{\mathrm{j}}$, are complex unknowns. An iterative numerical scheme for determining $\text{Im(p)}$ is developed and the roots $\text{p}$ as well as angular stress and displacement distributions are calculated and plotted for various material combinations. Comparison with exact solutions for the case of dissimilar Isotropie solids indicates good accuracy of the numerical solution. The orthotropic nature of the materials is shown to have a significant effect on stress maximums.     

Plane-strain mixed-mode near-tip fields in elastic perfectly plastic solids under small-scale yielding conditions

Within the context of the small-strain approach, plane-strain mixed-mode near-tip fields of a stationary crack in an elastic perfectly plastic Mises solid under small-scale yielding conditions are examined by finite element methods. Steady-state stress fields in the immediate vicinity of the crack tip are obtained as the remote loading of the elastic K-field increases. Asymptotic crack-tip solutions consisting of constant stress sectors, centered fan sectors, and an elastic sector are then constructed accordingly. The asymptotic crack-tip stress solutions agree well with the numerical results for a whole spectrum of mixed-mode loadings. Our mixed-mode near-tip solution with an elastic sector differs from that of Saka et al. by one (plastic) constant stress sector situated between the elastic sector and the neighbouring fan sector. The effect of the existence of the elastic sector on the near-tip fields is discussed in the light of the computational results. The plastic mixity factor of the near-tip field is given as a function of the elastic mixity factor of the prescribed K-field. This function is well bounded by that of the perfectly plastic limit of the corresponding solutions for power-law hardening materials given by Shih. Some issues pertaining to the numerical procedures such as the implementation of the small-scale yielding assumption are also addressed.     

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

The higher-order asymptotic solution of a quasi-static steadily propagating mode-I crack under the plane strain condition in an elastic perfectly-plastic compressible material is studied. In order to statisfy the higher-order compatibility equation for the rate of deformation in the centered fan sector, the stress near the crack tip is expanded asymptotically as an irregular logarithmic power series. The higher order terms near the crack tip were successfully derived. These higher order solutions are distinctly different from those for a stationary crack. The present solution for a growing crack is a one-parameter near-tip field based on a characteristic length A, through which the influence of loading and crack geometry enter into the near-tip field. This feature is substantiated by the numerical solution obtained by A.G. Varias and C.F. Shih. Comparisons between the analytic solution and the numerical results are presented.Presented at the Far East Fracture Group (FEFG) International Symposium of Fracture and Strength of Solids, 4–7 July 1994 in Xi'an, China.     

Continuous near-tip fields for a dynamic crack propagating in a power-law elastic-plastic material

By use of the J ₂ flow theory and the rectangular components of field quantities, the near-tip asymptotic fields are studied for a dynamic mode-I crack propagating in an incompressible power-law elastic-plastic material under the plan strain conditions. Through assuming that the stress and strain components near a dynamic crack tip are of the same singularity, the present paper constructs with success the fully continuous dominant stress and strain fields. The angular variations of these fields are identical with those corresponding to the dynamic crack propagation in an elastic-perfectly plastic material (Leighton et al., 1987). The dynamic asymptotic field does not reduce to the quasi-static asymptotic field in the limit as the crack speed goes to zero. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.

     

Derivation of the near-tip stress and displacement fields for a moving crack without using complex functions

This note presents a mathematically simple approach to deriving the equations for the mixed-mode stress and displacement fields near the tip of a moving crack. Coordinate transformations allow the results to be derived without the use of complex functions.     

On the effect of the release of residual stresses due to near-tip microcracking

This article provides a comprehensive theoretical treatment of the interaction effects associated with the release of a general residual stress field as a result of the formation of microcracks in the vicinity of a main crack. The theoretical formulations are based upon the use of the complex potentials of Muskhelishvili and an appropriate superposition procedure. The induced stress intensity factor at the main crack is obtained in a series form and the leading order closed form solution was utilized to elucidate two interesting features of the study. The first is concerned with shielding and amplification effects associated with the release of a uniform residual stress field from a single microcrack. The second is concerned with shielding effects due to the formation of a microcracking zone in the vicinity of a stationary and a steadily growing main crack.The results for the case of a single microcrack reveal that, depending upon the location and orientation of the mirocrack, shielding and amplification effects may become prevalent. In the case of a microcracking zone, equivalence between the current interaction model and the contiuum mechanics approach was established based upon similar microcrack nucleation criteria.     

On the contribution of a microhole in the near-tip stress field to the J-integral

The J-integral analysis is performed for the interaction of a macrocrack with a microhole in a homogeneous plane of a linear elastic solid. A simple relation is obtained between three kinds of the J-integral induced by the macrocrack tip, the microhole and the remote stress field, respectively. The anti-shielding effect or the shielding effect of the microhole on the macrocrack can be confined easily from the J-integral induced from the microhole. This revised version was published online in August 2006 with corrections to the Cover Date.