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.     

Dynamic effects on the near crack-line fields for crack growth in an elastic perfectly-plastic solid

The dynamic effects on the near crack-line fields for steady-state tensile crack growth in an elastic perfectly-plastic solid are investigated under plane stress condition in this paper. In the plastic loading zone, the stresses and particle velocities near the crack-line are expanded in powers of the distance y to the crack line, with coefficients which depend on the distance of the moving crack tip. Substituting the expansions into the equations of motion, the Huber-Mises yield criterion and the Prandtl-Reuss flow rule yield a system of non-linear ordinary differential equations for the coefficients. This equation system is solved by using the approximate approach proposed by J.D. Achenbach and Z.L. Li. Finally, the crack growth criterion of critical strain is employed to determine the value of the remote elastic stress intensity factor K ₁that would be required for a crack growing steadily at a given Mach number. It is also shown in this paper that the steady-state dynamic solution yields the quasi-static solution as the speed of crack growth tends to zero.

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Résumé On étude les effets dynamiques qu'exercent les champs de contrainte au voisinage de la ligne de fissuration sous des conditions de tension stable dans un solide parfaitement élastique-plastique et pour un état plan de tension.Dans la zone de sollicitation plastique, les contraintes et vitesses élémentaires au voisinage de la ligne de fissuration se distribuent selon une puissance de la distance y à partir de cette ligne, avec des coefficients qui sont eux-mêmes dépendant de la distance à l'extrémité de la fissure en mouvement. En substituant les dilatations dans les équations de mouvement, le critère de plastification de Huber-Von Mises et la loi de'ecoulement de Prandtl-Reuss conduisent à un systéme d'équations différentielles ordinaires non linéaires pour déterminer ces coefficients.On résoud ce système d'équations grâce à une approach proposée par J.D. Achenbach et Z.L. Li.Enfin, le critère de déformation critique entraînant une croissance de la fissure est utilisé pour déterminer la valeur du facteur d'intensité des contraintes élastiques lointaines K ₁qui serait requis pour qu'une fissure croisse régulièrement à un nombre de Mach donné. On montre également dans l'étude qu'une solution dynamique sous conditions stables conduit à une solution quasi-statique lorsque la vitesse de propagation de la fissure tend vers zéro.

     

Resistance curves for crack growth under plane-stress conditions in an elastic perfectly-plastic material

A recently developed solution for the plastic strain, ε^P_y(x, t), on the crack line is used in conjunction with a critical strain criterion to construct curves for k_R(a) versus a, where a is the increase in crack length. Resistance curves have been computed for various values of the critical plastic strain. They show a monotonic increase of K_R(a) with increase in crack length, to a constant steady-state value.     

Unsteady growth of mode III crack in elastic perfectly-plastic material

In this paper the assembly of the near-tip fields given by J. R. Rice is completed for the mode III crack growing quasi-statically and unsteadily in elastic perfectly-plastic material. The obtained results provide a particular example for the general theoretical relations between the steady state and unsteady state crack growth. Further, the general expression of the rate of crack opening displacement is obtained, which is similar to one by J.R. Rice and co-workers for mode I crack growing in elastic perfectly-plastic material. The fracture criterion of the critical opening displacement at a prescribed distance behind the crack tip is discussed. As a result, the theoretical J-resistance curves are given.     

Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements

Triangular and prismatic quadratic isoparametric elements, formed by collapsing one side and placing the mid-side node near the crack tip at the quarter point, are shown to embody the (1/√r) singularity of elastic fracture mechanics and the (1/r) singularity of perfect plasticity. The procedure of performing the fracture analysis for the case of small scale yielding is discussed, and the finite element results are compared with theoretical results. The proposed elements have wide application in the fracture analysis of structures where ductile fracture is investigated. They permit a determination of the relationship between crack tip field parameters, loading, and geometry. And for a given fracture criterion can be applied to the prediction of fracture in structures such as pressure vessels under in service conditions.     

Fatigue crack growth in residual stress fields

A fatigue crack growth (FCG) model for specimens with well-characterized residual stress fields has been studied using experimental analysis and finite element (FE) modeling. The residual stress field was obtained using four point bending tests performed on 7050-T7451 aluminum alloy rectangular specimens and consecutively modeled using the FE method. The experimentally obtained residual stress fields were characterized using a digital image correlation technique and a slitting method, and a good agreement between the experimental residual stress fields and the stress field in the FE model was obtained. The FE FCG models were developed using a linear elastic model, a linear elastic model with crack closure and an elastic–plastic model with crack closure. The crack growth in the FE FCG model was predicted using Paris–Erdogan data obtained from the residual stress free samples, using the Harter T-method for interpolating between different baseline crack growth curves, and using the effective stress intensity factor range and stress ratio. The elastic–plastic model with crack closure effects provides results close to the experimental data for the FCG with positive applied stress ratios reproducing the FCG deceleration in the compressive zone of the residual stress field. However, in the case of a negative stress ratio all models with crack closure effects strongly underestimate the FCG rates, in which case a linear elastic model provides the best fit with the experimental data. The results demonstrate that the negative part of the stress cycle with a fully closed crack contributes to the driving force for the FCG and thus should be accounted for in the fatigue life estimates.     

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.

     

A griffith crack problem for an inhomogeneous elastic material

Summary This paper examines the problem of a Mode I crack in a nonhomogeneous elastic medium. It is assumed that the shear modulus varies exponentially with the coordinate perpendicular to the plane of the crack. The problem is reduced to a Fredholm integral equation and in terms of its solution the normal components of stress and displacement are described. Expressions are also derived for the stress intensity factor and the crack energy. The effect of the inhomogeneity is examined and comparisons made with the corresponding results for the homogeneous material.     

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.     

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.     

Penny-shaped crack in an infinite elastic cylinder bonded to an infinite elastic material surrounding it

This paper concerns with the state of stress in a long elastic cylinder, with a concentric penny-shaped crack, bonded to an infinite elastic medium. The crack is assumed to be opened by an internal pressure and that the plane of the crack is perpendicular to the axis of the cylinder. The elastic constants of the cylinder and the semi-infinite medium are assumed to be different. The problem is reduced to the solution of a Fredholm integral equation of the second kind. Closed form expressions are obtained for the stress-intensity factor and the crack energy. The integral equation is solved numerically and results are used to obtain the numerical values of the stress-intensity factor and the crack energy which are graphed.     

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 new solution for 3D crack extension based on linear elastic stress fields

A solution to the 3D stress field based on the maximum tangential stress (MTS) criterion is presented in this paper. The solution allows for the estimation of the critical crack plane, the direction of growth in terms of both twist and tilt angles and the equivalent crack driving force for a given mixed-mode loading condition. It also shows the graphical relationship between the three different stress intensities for a given driving force. Initial results have shown good correlation with experimental data obtained from literature.     

Plane strain crack problems for elastic materials with variable properties

Distributions of stress, strain and displacement occurring at the tip of a crack in a material with properties dependent on the type of loading are investigated for the conditions of plane strain in both far-field tensile and shear loads. The causes of the dependence of material properties on the type of external forces are the various inhomogeneities such as microcracks, pores, inclusions or reinforcing components in a material. The behaviour of these inhomogeneities depends substantially on the conditions of loading or deformation. Hence, the deformation properties of a material are not fixed intrinsic material characteristics that are invariant to the loading conditions, but rather the macroproperties of such materials are stress-state-dependent ones, and this effect becomes more noticeable as the volume content of the inhomogeneities increases. The asymptotic solutions of crack problems are obtained on the basis of proposed stress-strain relations describing not only the stress-state dependence of material properties, but the interrelation between the characteristics of volume and shear deformation as well. In a non-uniform stress state the primary macrohomogeneous material becomes an heterogeneous one. The use of the stress function is not effective for the solution of plane strain crack problems for the materials under consideration. Therefore, an approach based on the corresponding representation for the strains is used. It is shown that the commonly used suppositions of the symmetry or anti-symmetry in the stress distribution relative to the crack plane can not be accepted, since they do not allow all the boundary conditions to be satisfied. The opening of the crack surfaces in the case of far shear field is observed. The influence of stress-state sensitivity of material properties on the values of the stress intensity factor is more significant for tensile crack than for the crack in far shear field.     

Plane strain crack problems for elastic materials with variable properties

Distributions of stress, strain and displacement occurring at the tip of a crack in a material with properties dependent on the type of loading are investigated for the conditions of plane strain in both far-field tensile and shear loads. The causes of the dependence of material properties on the type of external forces are the various inhomogeneities such as microcracks, pores, inclusions or reinforcing components in a material. The behaviour of these inhomogeneities depends substantially on the conditions of loading or deformation. Hence, the deformation properties of a material are not fixed intrinsic material characteristics that are invariant to the loading conditions, but rather the macroproperties of such materials are stress-state-dependent ones, and this effect becomes more noticeable as the volume content of the inhomogeneities increases. The asymptotic solutions of crack problems are obtained on the basis of proposed stress-strain relations describing not only the stress-state dependence of material properties, but the interrelation between the characteristics of volume and shear deformation as well. In a non-uniform stress state the primary macrohomogeneous material becomes an heterogeneous one. The use of the stress function is not effective for the solution of plane strain crack problems for the materials under consideration. Therefore, an approach based on the corresponding representation for the strains is used. It is shown that the commonly used suppositions of the symmetry or anti-symmetry in the stress distribution relative to the crack plane can not be accepted, since they do not allow all the boundary conditions to be satisfied. The opening of the crack surfaces in the case of far shear field is observed. The influence of stress-state sensitivity of material properties on the values of the stress intensity factor is more significant for tensile crack than for the crack in far shear field.