Thermal stress singularities at tips of a crack in a semi-infinite medium under uniform heat flow

In the framework of plane thermoelastic problems is discussed the thermal stress field near the tips of an arbitrarily inclined crack in an isotropic semi-infinite medium with the thermally insulated edge surface under uniform heat flow. The crack is replaced by continuous distributions of quasi-Volterra dislocations corresponding to line heat sources and edge dislocations, and we obtain a set of simultaneous singular integral equations for dislocation density functions, whose solution is given in the forms of series in terms of Tchebycheff polynomials of the first kind. By means of this method, the thermal stress singularities at the crack tips are estimated exactly and the stress intensity factors can be readily evaluated. Numerical results are given for the particular case where the surface of the inclined crack is maintained at constant temperature and the heat supplied across the surface of the crack vanishes as a whole. The effects of the distance from the crack tip to the edge surface of the semi-infinite medium and the angle of inclination of the crack on the stress intensity factors and the initial direction of crack extension are shown graphically.     

The distribution of stress near the tip of a radial crack at the edge of a circular hole

A Mellin transform technique is used to find an integral equation the solution of which is related to the stress intensity factor and the formation energy of a crack at the edge of a circular hole in an infinite elastic solid. The biaxial loading case is considered in detail and numerical results given.     

On a transient thermal stress problem in an infinite circular cylinder with a penny-shaped crack

In the present paper, we deal with finding the stress intensity factors under transient thermal loading in an infinitely long circular cylinder containing a penny-shaped crack. Variations of the stress intensity factors with time, which will be closely related with crack extension, are illustrated in the figures.     

On the elasto-plastic stress concentration at a circular hole in an anisotropic sheet

Summary The classical problem of determining the stress concentration factor at a circular hole embedded in an infinite sheet subjected to remote uniform tension is investigated. A finite strain elasto-plastic deformation theory based on Hill's new anisotropic flow theory [7] is used. It is shown that the governing field equations can be reduced to a single first order differential equation from which the stress concentration factor is obtained by a standard numerical method. The solution covers the entire elasto-plastic range and is valid for any strain hardening function. Comparison with experimental results, for a few materials, shows good agreement.With a pure power hardening law and within the framework of small strain plasticity, our results agree with those obtained from a more general solution discovered by Budiansky [8].With 3 Figures     

Stress singularities in a periodically layered composite near interface crack tips

The paper deals with the asymptotic analysis of stresses near interface crack tips in the periodically two-layered elastic composites. The problem is investigated for the plane state of strain within the framework of the homogenized model with microlocal parameters. The angular dependence of stresses at the crack tip is presented for different mechanical and geometrical properties of the composite.     

Plane stress deformation zones at crack tips in polycarbonate

The plastic deformation produced crack tips in polycarbonate (PC) films stretched in tension, has been characterized by optical and transmission electron microscopy. An extensive and diffuse region of deformation is formed in unannealed specimens. Within this zone the ratio (v _f) of local film thickness to the (undeformed) thickness far away from the crack varies gradually both along and across the zone. The minimum ratio of 0.5 occurs at the crack tip. In contrast to this behaviour, films annealed for a short time just below the glass transition temperature T _g showed a highly localized response, the plastic strain being confined to a well-defined flame shaped deformation zone (DZ) ahead of the crack. Within most of this DZ, v _f is constant at 0.7, rising to 1 over a distance of 10 m at the zone tip, and falling to 0.5 over a distance of 4 m around the crack tip. Bi-refringence measurements show that a high degree of molecular orientation occurs within the zone. These experiments support the idea that an increase in the localization of the plastic strain response upon annealing below T _g is responsible for the embrittlement of PC by such heat treatment.     

Thermal stresses near tips of an insulated line crack in a semi-infinite medium under uniform heat flow

On the basis of the two-dimensional theory of thermoelasticity, the thermal stress field near the tips of a thermally insulated line crack in a semi-infinite medium under steady-state uniform heat flow is discussed. The crack is replaced by continuous distributions of sources of temperature discontinuity and edge dislocations. Then we obtain a set of simultaneous singular integral equations for density functions. The solution is given in the form of the product of the series of Tchebycheff polynomials of the first kind and their weight function. By means of this method, the thermal stress singularities at the crack tips are estimated exactly and the stress intensity factors can be readily evaluated. The effects of the distance from the crack tip to the bounding plane surface of the semi-infinite medium and the angle of inclination of the line crack on the stress intensity factors are shown graphically.     

Green's function for a crack emanating from a circular hole in an infinite sheet

Stress-intensity factors, K _I and K _II, are obtained for a point loaded crack emanating from a circular hole in an infinite plate. A series approach and the Muskhelishvili formulation in the two-dimensional theory of elasticity are used to derive the solution. The applicability of the solution is demonstrated by using it as a Green's function to obtain K _I and K _II in the case of (1) biaxial tension of an infinite plate and (2) bending of a wide strip.

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Résumé Les facteurs d'intensité de contrainte K _I et K _II sont obtenus dans le cas d'une fissure émanant d'un trou circulaire dans une plaque infinie et soumis à une charge ponctuelle. On utilise une approche par développement en série et une formulation de Muskhelishvili dans la théorie bi-dimensionnelle de l'élasticité afin de dériver une solution. La possibilité d'appliquer la solution est démontrée en l'utilisant comme une fonction de Green pour obtenir des valeurs de K _I et K _II dans la cas de: (1) une tension biaxiale sur une plaque infinie et (2) la flexion d'une bande de largeur suffisante.

     

Finite Fracture Mechanics crack initiation from a circular hole

The brittle crack initiation from a circular hole in an infinite slab under uniaxial remote tensile load is investigated. The analysis consists of two parts. The former is focused on the difference between symmetric and asymmetric crack propagation. Different criteria in the framework of the Theory of Critical Distances are implemented, and the potentiality of coupled Finite Fracture Mechanics (FFM) approaches is highlighted from a theoretical point of view. The latter presents the experimental results obtained by carrying out ad‐hoc tensile tests on polymethyl‐methacrylate (PMMA) and general‐purpose polystyrene (GPPS) notched samples and the related FFM investigation. It is shown that FFM predictions are in very good agreement with the experimental results for both tested materials.     

The practical evaluation of stress intensity factors at semi-infinite crack tips

The solution of crack problems in plane or antiplane elasticity can be reduced to the solution of a singular integral equation along the cracks. In this paper the Radau-Chebyshev method of numerical integration and solution of singular integral equations is modified, through a variable transformation, so as to become applicable to the numerical solution of singular integral equations along semi-infinite intervals, as happens in the case of semi-infinite cracks, and the direct determination of stress intensity factors at the crack tips. This technique presents considerable advantages over the analogous technique based on the Gauss-Hermite numerical integration rule. Finally, the method is applied to the problems of (i) a periodic array of parallel semi-infinite straight cracks in plane elasticity, (ii) a similar array of curvilinear cracks, (iii) a straight semi-infinite crack normal to a bimaterial interface in antiplane elasticity and (iv) a similar crack in plane elasticity; in all four applications appropriate geometry and loading conditions have been assumed. The convergence of the numerical results obtained for the stress intensity factors is seen to be very good.     

Interaction between a circular hole and an elastic-plastic crack under uniform shear load1

Summary The theory of continuously distributed screw dislications is used to study the interaction between an elastic-plastic crack and a circular cylindrical hole in an infinite elastic medium. This medium is under uniformly distributed shear load at infinity which produces anti-plane deformation. Exact solutions are obtained for the effect of the hole on the dislocation distribution function, the applied load-plastic zone relationship, the relative shear displacement along the crack and the plastic zone, and the crack-tip opening displacement.

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Wechselwirkung zwischen einem kreisförmigen Loch und einem elastisch-plastischen Riß zufolge gleichförmiger Schubbeanspruchung

Zusammenfassung Zur Untersuchung der Wechselwirkung zwischen einem elastisch-plastischen Riß und einem kreisförmigen zylindrischen Loch in einem unendlich ausgedehnten, elastischen Material wird die Theorie der stetig verteilten Schraubenversetzungen verwendet. Der Körper ist im Unendlichen durch eine gleichmäßig verteilte Schubbelastung beansprucht, die eine nichtebene Deformation hervorruft. Exakte Lösungen werden für den Effekt des Loches in der Versetzungsverteilungsfunktion, das Verhältnis aufgebrachte Last zur plastischen Zone, die relative Schubverformung entlang des Risses und der plastischen Zone und die Rißspitzenverschiebung erhalten.

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With 8 Figures

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This work was based upon part of a dissertation submitted by the first author in partial fulfillment of the requirements for the Ph.D. degree at The University of New Mexico, December 1974.     

Influence of local variations of yield strength on plastic zones at crack tips

The preponderance of existing literature which deals with plastic zones near crack tips considers materials and analyses in which the yield strength value is taken to be constant and uniform throughout the region under investigation, at least initially. Such an assumption is not easily defended if one is to consider crack initiation and propagation in real engineering components which may have been forged, cold worked, annealed, or cast during manufacture. The present work establishes analytically and experimentally the influence of one-dimensional gradients of yield strength on the size and shape of plastic zones in single edge notched (SEN) 7075-T6 aluminum specimens.     

Fracture angle and strain-energy-density-factor of a crack at hole at an arbitrary angle

For both the maximum stress criterion and strain-energy-density-factor (S) theory, fracture angle (the initial angle of crack growth) − θ_o is predicted by using opening and sliding mate stress intensity factors, k₁ and k₂. These theoretical predictions are consistent with experimental fracture angles.

For the S theory, the crack spreads in the negative θ_o-direction in a plane for which S is a minimum, S_min. This quantity was obtained analytically. The experimental data of the critical S (S_cr) on plexiglass fracture specimens remains essentially constant.     

On a crack problem in an infinite circular cylinder with a penny-shaped crack under a transient thermal loading

In this paper we deal with finding the stress intensity factors under the transient thermal loading in a circular cylinder with infinite length containing a penny-shaped crack. Variations of the stress intensity factors with time, which are closely related with the crack propagations, are obtained and illustrated in figures. From these figures we can obtain useful suggestions respecting crack propagation.

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Résumé On traite, dans cette étude, de la recherche des facteurs d'intensité de contraintes dans un cylindre circulaire de longueur infinie comportant une fissure en forme d'angle et soumise à sollicitation thermique en régime transitoire.On obtient, et on donne des valeurs à titre d'exemples pour les variations de facteur d'intensité des contraintes en fonction du temps, qui sont en relation étroite avec la propagation de la fissure. Ces valeurs conduisent à des suggestions utiles en ce qui regarde la propagation d'une fissure.

     

A remark on the direct numerical determination of stress intensity factors at crack tips

A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss-Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto-Chebyshev method for the numerical solution of the same class of equations.     

Analysis of a crack emanating from a circular hole in a loaded plane

The problem of a linear elastic loaded plane containing a circular hole with an arbitrary positioned edge crack is considered. The formulation is based on consideration of the dislocational pile-up in order to represent the crack, and on a fictious dislocation in the hole in order to satisfy single valued displacement requirement for the resulting stress potentials.The results are given in terms of the stress intensity factors at the crack tip for a variety of geometries. As the results indicate, at certain geometries there exists a critical crack length such that, below that value, the crack experiences an intensive growth, and, above this value, retardation of growth or crack arrest will take place.

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Résumé On considère le problème d'une plaque plane linéaire élastique, mise en charge, et comportant un trou circulaire avec une fissure de bord placée de manière arbitraire. Pour la formulation, on se base sur une représentation de la fissure par un empilement de dislocations et, pour satisfaire à l'exigence de déplacement unitaire associé aux potentiels de contrainte résultants, on suppose l'existence d'une dislocation fictive dans le trou.On présente les résultats sous forme de facteurs d'intensité de contraintes correspondant à diverses géométries. Comme l'indiquent les résultats, il existe pour certaines configurations une longueur critique de fissure telle que la fissure passe par une phase de croissance intensive lorsque sa longueur en est inférieure, ou au contraire se ralentit, voire s'arrête, lorsque sa longueur est supérieure à cette longueur critique.

     

The stress intensity factor of a crack emanating from a circular hole in a finite plate by boundary collocation method

The problem of a finite plate containing a circular hole with a crack subjected to a uniform remote stress is analysed by boundary collocation method. For an infinite plate with dimensions large in comparison with the crack length and the radius of the hole, the stress intensity factor (SIF) values calculated by this method coincide very well with the existing solution. Numerical results of the SIF values are obtained for varying crack-length-to-hole-radius ratios, l/r, and equivalent crack-length-to-plate-width ratios, a/b. The analysis for finite plates can provide useful results for practical applications.

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Résumé On analyse par la méthode de collocation des limites le problème d'une plaque finie comportant un trou circulaire avec une fissure, soumis à une contrainte uniforme appliquée à une certaine distance. Dans le cas d'une plaque infinie et de grandes dimensions par rapport à la longueur de la fissure et le rayon du trou, les valeurs du facteur d'intensité de contraintes calculées à l'aide de cette méthode coincident bien avec la solution existante. On obtient des résultats numériques du facteur d'intensité de contrainte pour différents rapports l/r de la fissure sur le rayon du trou, et pour les rapports a/b de la longueur de fissure équivalente sur la largeur de la plaque. L'analyse relative à des plaques finies peut fournir des résultats utiles à des applications pratiques.