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.     

Near-tip field of an anti-plane crack in a micro-cracked power-law solid

Asymptotic near-tip field is investigated for an anti-plane (mode III) crack in a power-law solid permeated by a distribution of micro-cracks. The micro-crack location is assumed to be random, while the micro-crack orientation is taken to be non-random. The anisotropic nature of this kind of damage gives rise to anisotropic constitutive equations for the overall macroscopic strains and stresses. The structure of the asymptotic field at a macro-crack tip is analyzed by solving a nonlinear eigenvalue problem. It is shown that under the assumptions made in this analysis the asymptotic crack tip field of the damaged solid has the same structure as the mode III HRR-field of the undamaged solid. Numerical results are presented for the angular functions, the contours of constant effective shear stress, the normalization constant arising in the near-tip field, and the crack opening displacement. By means of these results, the effects of the micro-crack density and orientation on the crack-tip field will be explored.     

Finite deformation finite element analyses of tensile growing crack fields in elastic-plastic material

Finite deformation finite element analyses of plane strain stationary and quasi-statically growing crack fields in fully incompressible elastic-ideally plastic material are reported for small-scale yielding conditions. A principal goal is to determine the differences between solutions of rigorous finite deformation formulation and those of the usual small-displacement-gradient formulation, and thereby assess the validity of the (nearly all) extant studies of ductile crack growth that are based on a small-displacement-gradient formulation. The stationary crack case with a significantly blunted tip is studied first; excellent agreement in stress characteristics at all angles about the crack tip and up to a radius of about three times the crack tip opening displacement is shown between Rice and Johnson's [1] approximate analytical solution and our numerical solution. Outside this radius, the numerical results agree very well with Drugan and Chen's [2] small-displacement-gradient analytical characteristics solution in the region of principal plastic deformation. Thus we identify accurate analytical representations for the stress field throughout the plastic zone of a blunted stationary crack. For the growing crack case, the macroscopic difference in crack tip opening profiles between previous small-displacement-gradient solutions and the present results is shown to be negligible, as is the difference in the stress fields in plastic regions. The stress characteristics again agree very well with analytical results of [2]. The numerical results suggest—in agreement with a recent analytical finite deformation study by Reid and Drugan [3]—that it is the finite geometry changes rather than the additional spin terms in the objective constitutive equation that cause any differences between the small-displacement-gradient and the finite deformation solutions, and that such differences are nearly indistinguishable for growing cracks.     

Numerical simulations of specimen size and mismatch effects in ductile crack growth - Part II: Near-tip stress fields

The effect of ductile crack growth on the near tip stress field in two different specimen geometries has been investigated. For homogeneous specimens it is observed that the peak stress level increases with ductile crack growth. The effect is most pronounced up to about 1 mm of crack growth. For low and intermediate hardening there is a significant effect of specimen size on the stress level. In case of mismatch in yield stress, the simulations show that the increase in stress level in the material with the lower yield stress is of a similar magnitude as is the case for stationary cracks. In case of ductile crack growth deviation from the original crack plane occurs, the highest stresses are still found close to the interface, and not in front of the current crack tip.     

Crack-tip fields in elastic-plastic material under plane stress mode I loading

New results on the crack-tip fields in an elastic power-law hardening material under plane stress mode I loading are presented. Using a generalized asymptotic expansion of the stress function, higher-order terms are found which have newly-discovered characteristics. A series solution is obtained for the elastic-plastic crack-tip fields. The expansion of stress fields contains both the and terms where t_i is real and t_k is complex; the terms σ⁽ⁱ⁾ _pq(θt_i) and σ^(k) _rsθt_k) are real and complex functions of θ respectively. Comparing the results with that for the plane strain mode I loading shows that: (1) the effect of higher-order solutions on the crack-tip fields is much smaller; and (2) the path-independent integral J also controls the second-order or third-order term in the asymptotic solutions of the crack-tip fields for most of the engineering materials (1 < n < 11) in plane stress, while the J-integral does not control the second and the third-order terms for the plane strain mode I case for n > 3. These theoretical results imply that the crack-tip fields can be well characterized by the J-integral, and can be used as a criterion for fracture initiation under plane stress mode I loading. This is in agreement with existing full-field solutions and experimental data that J at crack growth initiation is essentially independent of in-plane specimen geometry. The comparison confirms the theoretical asymptotic solutions developed in this study. This revised version was published online in August 2006 with corrections to the Cover Date.     

A perturbation solution for a crack in a power-law material under gross yielding

Abstract In order to develop an analytical method for quantifying the plastic-blunting behaviour of a short crack embedded in a notch plastic zone, the perturbation solution of He and Hutchinson is extended to include the effect of strain gradient. An edge-cracked plate subjected to a linearly varying remote strain is considered in this work to simulate the plastic deformation associated with a small crack at a notch root. The strain hardening of the material is assumed to obey a power-law. Comparison with finite-element (FE) computations demonstrates that this perturbation solution provides accurate values for the crack-tip opening displacement (CTOD) under gross-yielding conditions for a range of hardening parameters.     

Stability of a crack in a slightly work hardening elastic-plastic material

The stability is considered of a crack in a slightly work hardening elastic-plastic material under a uniform normal applied stress at infinity.

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Zusammenfassung Die Stabilität eines Risses in einem durch Behandlung nur leicht härtbaren elastischplastischen Material unter dem Einfluss einer gleichförmig zunehmenden, senkrecht wirkenden Belastung wird behandelt.

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Résumé On envisage les conditions de stabilité d'une fissure dans un material elasto-plastique legérement sensible au durcrissement par écrouissage, soumis a une tension normale uniforme appliquée à l'infini.

     

Creep crack growth in an elastic-creeping material Part I: mode III

The quasi-static growth of a crack in an elastic-creeping material under antiplane shear or mode III loading is investigated. The creep response of the material is assumed to be governed by a power-law between the creep strain rate, creep strain, and stress. While this law is capable of describing elastic-primary, secondary, or tertiary creep, the major emphasis of this paper is on crack growth in the elastic-primary creep regime. The asymptotic crack-tip stress and strain fields for a quasi-statically extending crack in an elastic-primary creeping material are developed. This is followed by a finite element analysis to determine the complete stress and strain fields within the confines of small scale yielding. These fields are then compared with the asymptotic ones to establish the size of the zone of dominance of the crack-tip fields.

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Résumé On étudie la croissance quasi-statique d'une fissure dans un matériau en phase de fluage élastique sous des cisaillements anti-planaires ou sous une sollicitation de mode III.La réponse du matériau sur fluage est supposée gouvernée par une loi parabolique qui lie la vitesse de déformation par fluage, la déformation par fluage et la contrainte. Si cette loi est à même de décrire les fluages élastiques-primaire, secondaire et tertiaire-l'accent est ici principalement placé sur la croissance d'une fissure sous un fluage élastique-primaire. On établit les champs de contrainte et de déformation asymptotiques à l'extrémité de la fissure dans le cas d'une fissure en croissance quasi-statique, dans un matériau soumis à fluage élastique primaire.On procède ensuite à une analyse par éléments finis afin de déterminer complètement les champs de contraintes et de déformation aux confins d'une zone de petite taille en déformation plastique. On compare ces champs aux champs asymptotiques, en vue d'établir la taille de la zone où les champs à l'extrémité de la fissure s'exercent de manière dominante.

     

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 asymptotic crack tip fields for a crack perpendicular to an interface between elastic-plastic materials

Summary The singular behavior near a crack tip at the interface between two power-law hardening materials with the crack perpendicular to the interface is studied for both Mode I and Mode II loading under either plane strain or plane stress conditions. The mathematical model developed can be expressed as a fourth order ordinary differential equation with homogeneous boundary condition. A shooting method is applied to obtain the eigenvalues and to solve the differential equation with homogeneous boundary conditions. When both materials have the same hardening exponent,N, another material parameter, , representing the relative resistance of two materials to plastic deformation, is introduced to reflect the joint effect of the two materials on the singularity. Results indicate that if both materials have the sameN, the singularity at the crack tip is reduced as increases; however, when becomes large there appears to be little change in the singularity for a fixedN. When the hardening exponents are not the same, the mathematical model assumes stress continuity across the interface. The results show that the order of the singularity depends largely on the softer material, with the largest stresses in the harder material.     

Creep crack growth in an elastic-creeping material Part II: mode I

The quasi-static growth of a crack in an elastic-creeping material under mode I loading is investigated. The creep strain rate of the material is assumed to be governed by a power law involving the stress and creep strain. The major emphasis of this investigation is on elastic-primary creep response. The asymptotic crack tip fields for a quasi-statically extending crack under conditions of plane strain and plane stress are developed. The asymptotic fields are unambiguously determined in terms of the instantaneous crack speed and material parameters and are independent of the prior crack history, specimen geometry, and loading. A plane strain finite element analysis is performed to determine the complete stress and strain fields. These fields are compared with the asymptotic ones to establish the zone of dominance of the crack tip fields. The zone of dominance can be a very small fraction of the size of the creep zone attending the crack up.

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Résumé On étudie la croissance quasi-statique d'une fissure dans un matériau sujet à fluage élastique sous une sollicitation de Mode I. On suppose que la vitesse de déformation en fluage du matériau est régie par une loi parabolique comportant la contrainte et la déformation. L'accent est surtout mis sur la réponse au fluage élastique primaire.On établit les configurations asymptotiques règnant à l'extrémité de la fissure, dans le cas d'une fissure quasi-statique en extension sour état plan de dilatation ou sous état plan de tension. Ces champs sont déterminés sans ambiguïté par la vitesse instantanée de la fissuration et par les paramétres du matériau; ils sont indépendants de l'histoire primitive de la fissure, de la géométrie de l'éprouvette et de la sollicitation. On effectue une analyse par éléments finis en déformations planes pour déterminer complètement les champs de contrainte et de dilatation, que l'on compare aux champs asymptotiques aux fins d'établir la zone de prédominance des champs à l'extrémité de la fissure. Cette zone peut être une fraction très petite de la taille de la zone où s'effectue un fluage au voisinage de l'extrémité de la fissure.

     

Composite crack profile model for CTOD determination III: An experimental application to elastic-plastic crack growth situations

A composite crack profile (CCP) model has been applied for the evaluation of CTOD in the elastic-plastic crack growth situations prevailing in a structural steel. The results have been compared with the ones obtained by conventional method (using plastic hinge model such as Wells etc.) The CTOD-Resistance Curves (δ_R-curves) have also been obtained as a function of specimen thickness, a/w ratio and the loading geometry by using the CCP model. The significance of crack initiation CTOD (δ_i) and the maximum load CTOD (δ_m) has been discussed in relation to various geometrical parameters (i.e. thickness, a/w ratio and loading geometry).     

The mechanics of a constantly growing crack in an elastic power-law creeping material

The mechanics of transient crack growth in an elastic power-law creeping material is investigated using the special case of a suddenly growing crack with constant growth rate \.a ₀. Small scale yielding assumption is assumed to be valid at the instant when crack growth occurs. The analysis is carried out for the case of antiplane shear mode III and plane strain mode I. The resulting transient stress field for small crack extension is analyzed using perturbation methods. For small crack extension, the HR field, the RR field and the elastic K field coexist near the crack tip, one inside another. The regions of dominance of these near tip fields are estimated and an approximate solution is provided for the near tip stresses. The effect of crack growth rate on the small scale yielding condition is also studied. For large crack extension, it is shown that creep relaxation can be neglected and results of steady state analysis can be modified to describe the near tip stress fields. Extension of these results to more general crack growth histories is also discussed

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Résumé On étudie le mécanisme de la croissance d'une fissure en phase transitoire dans un matériau élastique satisfaisant une loi de fluage de forme parabolique, en considérant le cas particulier d'une fissure en croissance soudaine suivant une vitesse de croissance constante a₀.On suppose valide l'hypothèse de la déformation plastique à petite échelle, au moment où la fissure s'étend. L'analyse est effectuée pour un cisaillement antiplanaire (Mode 3) et pour un état plan de déformation (Mode 1). En utilisant la méthode des perturbations, on calcule le champ de tensions transitoires associé à une petite extension de la fissure. Pour ce cas, le champ HR, le champ RR et le champ élastique K coexistent, l'un dans l'autre, près de l'extrémité de la fissure. On évalue les régions où dominent ces champs et on propose une solution apportée pour les contraintes au voisinage de l'extrémité de la fissure. On étudie également l'effet de la vitesse de croissance de la fissure sur la condition d'écoulement plastique à petite échelle. Pour de grandes extensions de fissure, on montre que la relaxation par fluage peut être négligée, et que les résultats de l'analyse de régime stable peuvent être modifiés pour décrire les champs de contrainte au voisinage de l'extrémité de la fissure. On discute également l'extension de ces résultats à des histoires de croissance de fissure de caractère plus général

     

An analysis of antiplane shear crack growth in a rate sensitive elastic-plastic material

The problem of steady growth of an antiplane shear crack in a strain rate sensitive elastic-plastic material is considered. It is assumed that the material is elastic-perfectly plastic under slow loading conditions, but that the inelastic strain rate is proportional to the difference between the stress and the yield stress raised to some power. In earlier work on this problem in which the possibility of an elastic region in stress space was not considered, it was concluded that the asymptotic crack tip field is completely autonomous, but the local field could not be shown to reduce to a generally accepted rate independent limit as the rate sensitivity of the material vanished. It is shown here that, if the possibility of elastic unloading is admitted in the formulation, the asymptotic crack tip solution does indeed approach the correct rate independent limit as the rate sensitivity of the material vanishes. Furthermore, the crack tip field loses the feature of complete autonomy under these conditions. That is, the crack tip field involves an undetermined parameter that can be determined only from the remote fields. In the analysis, both dynamic and quasistatic growth are considered.

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Résumé On considère le problème de la croissance stable d'une fissure de cisaillement antiplanaire dans un matériau élastoplastique sensible à la vitesse de déformation. On suppose que le matériau est élastique puis parfaitement plastique en charge lente, mais que la vitesse de déformation inélastique est proportionnelle à la différence entre la contrainte appliquée et la limite élastique, à une certaine puissance. Dans un travail antérieur sur ce problème, où l'on n'avait pas envisagé une région élastique dans l'espace de sollicitations, on avait conclu que le champ asymptotique correspondant à l'extrémité d'une fissure était complètement autonome. Toutefois, on ne pouvait établir que le champ local se réduise à une limite généralement admise comme indépendant de la vitesse, lorsque disparaissait la sensibilité du matériau à la vitesse de sollicitation. Dans la présente étude, on montre qui si on admet dans la formulation la possibilité d'un déchargement en régime élastique, la solution du champ asymptotique à l'extrémité de la fissure se rapproche vraiment de la limite correcte d'indépendance à la vitesse, lorsque disparait la sensibilité du matériau à cette vitesse. En outre, le champ à l'extrémité de la fissure perd ses caractéristiques de complète autonomie sous ces conditions. Ceci veut dire qu'il comporte un paramètre indéterminé, qui ne peut être déterminé qu'à partir des champs de contrainte plus lointains. Dans l'analyses, on a considéré à la fois la croissance dynamique et quasistatique d'une fissure.

     

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.     

On the dependence of the dynamic crack tip temperature fields in metals upon crack tip velocity and material parameters

Although various approximations have been used to analytically predict the temperature rise at a dynamic crack tip and its relation to the crack tip velocity or the material properties, few experimental investigations of these effects exist. Here, the method of using a high speed infrared detector array to measure the temperature distribution at the tip of a dynamically propagating crack tip is outlined, and the results from a number of experiments on different metal alloys are reviewed. First the effect of crack tip velocity in 4340 steel is investigated, and it is seen that the maximum temperature increases with increasing velocity, the maximum plastic work rate density increases with velocity and the active plastic zone size decreases with increasing velocity. Also, it is observed that a significant change in the geometry of the temperature distribution occurs at higher velocities in steel due to the opening of the crack faces behind the crack tip. Next, the effect of thermal properties is examined, and it is seen that, due to adiabatic conditions at the crack tip, changes in thermal conductivity do not significantly affect the temperature field. Changes in density and heat capacity (as well as material dynamic fracture toughness) are more likely to produce significant differences in temperature than changes in thermal conductivity. Finally, the effect of heat upon the crack tip deformation is reviewed, and it is seen that the generation of heat at the crack tip in steel leads to the localization of deformation in the shear lip. The shear lip is actualy an adiabatic shear band formed at 45° to the surface of the specimen. In titanium, no conclusive evidence of shear localization in the shear lip is seen.     

A new form of exact solutions for mode I,II, III crack problems and implications

A new form of an exact linear elastic solution is obtained for the problem of a crack in a semi-infinite plate subjected, at infinity, to antiplane stress (Mode III) loading. The use of the conformal mapping technique results in a convenient stress and displacement solution for each point of the semi-infinite domain. For completeness, the solution technique is extended to Mode I, II problems with center-cracks as well as the Mode III V-notch problem. It is shown that the limiting case of the V-notch collapses to the stress functions independently derived for the edge-cut. At distances equivalent to 10% of the crack length away from the crack tip, the exact solutions give stresses about 7.5% greater than the one-term results. Ramifications of the exact solutions to finite-element solutions, elastic-plastic and diffusion problems are discussed.     

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid are investigated under plane strain and plane stress conditions. In each case, by expanding the plastic fields and the governing equations in the coordinate y, the problem is reduced to solving a system of nonlinear ordinary differential equations which is similar to that of mode III derived by Achenbach and Z.L.Li. An approximate solution for small values of x is obtained and matched with the elastic field of a blunt crack at the elastic-plastic boundary. The crack growth criterion of critical strain is employed to determine the value of K _II of the far-field that would be required for a steadily growing crack.     

Mode I crack in a soft ferromagnetic material

ABSTRACT In the existing magnetoelastic theories, stress is proportional to the square of magnetic intensity and the linear model developed is usually used to analyse magnetoelastic problems. For a crack problem, the perturbation of the magnetic field caused by deformation is not much less than the applied field. In this paper, complex potentials for a mode I crack with a nonlinear relation for magnetic intensity are developed. The boundary conditions on crack faces are represented in terms of the continuity of the magnetic field. A solution of the crack problem is obtained by solving the Riemann‐Hilbert problem. Making use of the solution, the effects of the boundary conditions on the crack faces on the magnetoelastic coupling are discussed.     

A perturbation analysis of combined mode I and III dynamic crack propagation

Summary In the present paper the asymptotic stress and deformation fields of dynamic crack extension in materials with linear plastic hardening under combined mode I (plane strain and plane stress) and anti-plane shear loading conditions (mode III) are investigated. The governing equations of the asymptotic crack-tip fields are formulated from two groups of angular functions, one for the in-plane mode and the other for the anti-plane shear mode. It was assumed that all stresses and deformations are of separable functional forms ofr and , which represent the polar coordinates centered at the actual crack tip. Perturbation solutions of the governing equations were obtained. The singularity behavior and the angular functions of the crack-tip in-plane and the anti-plane stresses obtained from the perturbation analysis show that, regardless of the mixity of the crack-tip field and the strain-hardening, the in-plane stresses under the combined mode I and mode III conditions have stronger singularity in the whole mixed mode steady-state crack growth than that of the anti-plane shear stresses. The anti-plane shear stresses perturbed from the plane strain mode I solutions lose their singularity for small strain hardening, whereas the angular stress functions perturbed from the plane stress mode I have a nearly analogous uniform distribution feature compared to pure mode III cases. An obvious deviation from the unperturbed solution is generally to be observed under combined plane strain mode I and anti-plane mode III conditions, especially for a large Mach number in a material with small strain-hardening; but not under plane stress and mode III conditions. The crack propagation velocity decreases the singularities of both pure mode and perturbed crack-tip fields.