Influence of load biaxiality on the fracture characteristics of two collinear cracks

A theoretical investigation has been performed to determine the influence of load biaxiality on quantities pertinent to the brittle (elastic) fracture of an infinite sheet containing a pair of collinear cracks. It is shown how the biaxiality of the applied load affects the stress, the displacements and the maximum shear near the ends of the cracks, as well as the angle of initial crack extension and the displacement of the crack borders. The influence of load biaxiality on the rate at which the elastic strain energy of the entire body is altered with change in the length of the cracks is also demonstrated, and is used in conjunction with Griffith's crack instability hypothesis to show the effect of load biaxiality on the fracture load. The analysis indicates that loads applied parallel to the cracks influence the value of the critical (fracture) tensile load applied perpendicular to them, and that the Poisson ratio of the material determines the characteristics of this influence.

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Résumé On a procédé à une étude théorique en vue de déterminer l'influence de la biaxialité de la mise en charge sur les paramètres qui déterminent la rupture fragile (élastique) dans une feuille infinie comportant une paire de fissures colinéaires. On montre comment la biaxialité de charge appliquée affecte la contrainte, les déplacements et le cisaillement maximum au voisinage des extrémités des fissures, de même que l'angle de l'extension de fissure initiale et que le déplacement des bords de la fissure. L'influence de la biaxialité des charges sur la vitesse à laquelle l'énergie de déformation élastique de l'ensemble du corps se trouve altéré par le changement de longueur des fissures, est également démontrée. En combinaison avec l'hypothèse d'instabilité de fissure de Griffith, on fait usage de cette influence pour montrer l'effet de la biaxialité de contrainte sur la charge de ruine. L'analyse indique que les contraintes appliquées parallèlement aux fissures influencent la valeur de la charge critique de traction conduisant à rupture appliquée perpendiculairement à celles-ci, et que le module de Poisson du matériau détermine les caractéristiques de cette influence.

     

The influence of biaxiality of loading on the form of caustics in cracked plates

Summary The optical method of reflected caustics was applied up-to-now to problems of cracked plates under uniaxial loading. Only the problem of the biaxial tension of the plate has been considered for the particular case where the crack is transverse to the longitudinal axis of the plate which coincided with the loading axis. In this paper the influence of a biaxial loading of the plate on the form and orientation of the caustic was studied in connection with the orientation of the crack. New modified relations were given for the evaluation of the complex stress intensity factorK=K _I –iK _II in terms of the angle of the angular displacement of the caustic axis. For the accurate evaluation ofK _I andK _II nomograms of correction factors _y ^max , _x ^max and _x ^max were given in terms of the angle of inclination of the crack =(90–) and the biaxiality factork. Experimental evidence with PMMA internally cracked plates corroborated the results of theory.List of Symbols (z), (z) complex-stress function of Muskhelishvili - _xx , _yy , _xy crack tip stress referred to Cartesian coordinate system - r, polar coordinate system centered at crack tip - K _I ,K _II stress intensity factors for ModeI andII loading, respectively - angle of inclination of the crack - 90°- - k ratio of stresses at infinity - ₁, ₂ principal stresses at crack tip - a crack length - stress applied at infinity along the transverse boundaries of the plate - X _r,f ,Y _r,f parametric equations of the reflected caustics referred to the Cartesian systemOXY on the reference screen: (r) reflected caustics from rear face of the specimen and (f) reflected caustics from the front face of the specimen - r ₀ radius of the generatrix curve on the specimen around the crack tip (initial curve) - c _r,f optical constants of the material for reflections from the rear and front faces of the specimen respectively - _m magnification ratio of the optical set-up - z ₀ distance between the reference-screen and the middle plane of the specimen - z _i distance between the focus of the light beam and the middle plane of the specimen - d thickness of specimen - 2 for the reflected caustics from the rear face of the specimen and 1 for the reflected caustics from the front face of the specimen - C _r,f z ₀ dc _r,f / _m (2)^1/2 - v Poisson's ratio - E elastic modulus of the material - A (1+k)+(1–k) cos 2 - B (1–k) sin 2 - C 1+k ²+(1–k ²) cos 2 2 tan^–1 (B/A)=2 tan^–1 (K _II/K_I) - D _y ^max ,D _x ^max ,D _x ^min the maximum and the ninimum diameter of caustics along the axisOy andOx of the crack respectively - _y ^max , _x ^max , _x ^min the correction factors forD _y ^max ,D _x ^max andD _x ^min respectively - D _t ^max ,D _l ^max the maximum transverse and longitudinal diameters of the caustics respectively - _t ^max , _l ^max the correction factors forD _t ^max ,D _l ^max respectively With 13 Figures     

On the effect of initial curvature on cracked flat sheets

A survey of existing solutions describing the stress distribution around the crack tip of an initially curved sheet is made and a method for estimating approximate stress intensity factors of other more complicated shell geometries is discussed.

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Zusammenfassung Der Verfasser gibt cine Übersicht der Lösungen, welche die Spannungsverteilung in der Umgebung der Spitze eines Risses in einem vorgebogenen Blech beschreiben. Außerdem wird ein Verfahren besprochen, nach dem Spannungsintensitätsfaktoren im Falle komplizierterer geometrischer Formen näherungsweise geschätzt werden können.

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Résumé Il est procédé à une Etude des solutions existantes qui permettent de décrire la distribution des tensions au voisinage de l'extrémité d'une fissure dans une tôle mince présentant une courbure donnée.L'on discute également une méthode d'estimation approchée des facteurs d'intensité de contrainte dans le cas d'enveloppes à géométrie plus complexe.

     

Influence of anisotropy on the limit load of bi-material welded cracked joints subject to tension

A plane-strain upper bound limit load solution for bi-material welded joints subject tension is found. It is assumed that each material obeys Hill's orthotropic yield criterion and one of the principal axes of orthotropy coincides with the tensile direction. A crack of arbitrary length is located at the bi-material interface. The solution is based on a simple discontinuous kinematically admissible velocity field and is an extension of the corresponding solution for the specimen made of isotropic materials. The main purpose of the paper is to demonstrate that the influence of anisotropy on the magnitude of limit loads may be much more significant than other effects.     

Plastic J-integral calculations using the load separation method for the center cracked tension specimen

The load separation method was used to determine the plastic work factor (η_pl) for the center crack tension geometry for power law hardening materials with Ramberg-Osgood hardening exponents n ranging from 2 to 20 and crack sizes a/W ranging from 0.4 to 0.8. The resulting expression for η_pl compares favorably to the analytical work of Rice, Paris, and Merkle and to Sharobeam and Landes. The results were also compared to η_pl calculated from the plastic J results in EPRI Handbook NP-1931. Unlike the EPRI results, η_pl from the load separation method are not a function of crack size.     

Influence of anisotropy on a limit load of weld strength overmatched middle cracked tension specimens

ABSTRACT A plane‐strain upper bound limit load solution for weld strength overmatched middle cracked tension specimens (M(T) specimens), is found. It is assumed that the weld material is isotropic, but the base material is orthotropic and its axes of orthotropy are straight and parallel to the axes of symmetry of the specimen. A quadratic orthotropic yield criterion is adopted. The solution is based on a simple discontinuous kinematically admissible velocity field and is an extension of the corresponding solution for the specimen made of isotropic materials. These two solutions are compared to demonstrate the influence of anisotropy on the magnitude of the limit load.     

The effect of stress biaxiality on fracture initiation at a blunt flaw

From the basis of the cohesive process zone representation of the micro-mechanistic processes that are associated with fracture, the author has, in earlier work, derived a blunt flaw fracture initiation criterion for situations where the process zone size s is small compared with the flaw depth (length) and any other characteristic length, e.g. remaining ligament width, of a configuration, other than the flaw root radius ρ. The criterion gives the critical elastic flaw tip peak stress σ_pcr in terms of the process zone material parameters and geometrical parameters, and was derived using a two-extremes procedure, whereby the separate and exact solutions for small and large s/ρ values are blended together to give an all-embracing relation that is valid for all s/ρ values. For a wide range of flaw geometry parameters and loading scenarios, it has been demonstrated that σ_pcr approximately depends on only one geometrical parameter: ρ. In this paper, we show that in the presence of transverse stresses in the direction of the flaw, i.e. with biaxial stress situations, σ_pcr depends, in a coupled manner, on the flaw shape and the degree of stress biaxiality.     

Influence of test parameters on the measurement of the essential work of fracture of zinc sheets

The Essential Work of Fracture (EWF) concept is used to characterize the fracture of thin plates of a zinc alloy. The consistency and applicability of the EWF approach are discussed. The successive stages of the fracture process were studied: the evolution of the shape of the crack tip was observed with a scanning electron microscope, the shape of the plastic zone and the evolution of necking in the ligament ahead of the crack tip were measured using a laser profilometer and the onset of cracking in the ligament was detected by means of a TV camera coupled with acoustic emission recording. The influence of both test parameters and material parameters on the Essential Work of Fracture (W _e ) and on the CTOD (_c) are elucidated. The effects of specimen geometry, deformation rate, texture and grain size are especially investigated. The main advantages and drawbacks of the EWF method are highlighted.     

Dugdale yielded zones in cracked sheets of glassy polymers

Photoelastic measurements on centrally notched sheets of polycarbonate have been used to confirm the elastic stress field around the yielded zone predicted by the Dugdale model. By comparing the behaviour of polycarbonate, polysulfone and Polyvinylchloride the plasticity requirements of the model have been investigated. The residual elastic stress fields in unloaded specimens have been interpreted in terms of a lower compressive than tensile yield stress.     

Influence of local fracture energy distribution on maximum fracture load of three-point-bending notched concrete beams

The maximum fracture load of a notched concrete beam has been related to the local fracture energy at the cohesive crack tip region analytically in this paper, and then the correlation between the size effects on the maximum fracture loads and the RILEM specific fracture energy is established. Two extreme conditions have been established, namely zero crack-tip bridging with zero local fracture energy and maximum crack-tip bridging with the maximum size-independent fracture energy. It is concluded that the local fracture energy at the crack tip region indeed varies with the initial crack length and the size of specimen. The tri-linear model for the local fracture energy distribution is confirmed by using the proposed simple analytical solution.     

On the use of special crack tip elements in cracked elastic sheets

This paper presents a finite element method for determining the stress intensity factor in a cracked elastic sheet. Special cracked elements are placed around each crack tip; in each special element the stresses and displacements are derived from the exact stress function while the continuity of the displacements at the nodes is satisfied in a least square sense. A general procedure for evaluating the stiffness matrix of a cracked isotropic, or orthotropic, element is presented, and the numerical results obtained are compared with exact analytical results.

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Résumé Le mémoire présente une méthode par éléments finis pour déterminer le facteur d'intensité de contrainte dans un feuillard élastique fissuré. Des éléments spéciaux, présentant des fissures, sont disposés autour de chaque extrémité de fissures. Dans chacun de ces éléments spéciaux, les contraintes et les déplacements sont déduits de la fonction exacte d'une contrainte tandis que la continuité des déplacements aux noeuds satisfait à une règle des moindres carrés. Une procédure générale pour l'évaluation de la matrice de rigidité d'un élément fissuré à caractère isotrope ou orthotrope est présentée, et les résultats numériques obtenus sont comparés avec les résultats analytiques exacts.

     

General criterion of fracture of cracked plates

International Journal of Fracture -     

Combined mode fracture via the cracked Brazilian disk test

An analysis is made of the cracked Brazilian disk test. Explicit results are given for the stress intensity factors of a crack aligned at any direction to the applied load. Various approximations are considered and a simplified short crack approximation is shown to be accurate for /a0.3 ( is crack length, a the disk radius). At certain angles it is found that crack closure should occur and as a frist step the effect of friction is taken into account for such cases by using this short crack approximation. Finally, experimental results are presented to confirm and elucidate the use of this specimen geometry for the problem of combined mode fracture.

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Résumé On a procédé à une analyse de l'essai du disque brésilien fissuré. Des résultats explicites sont fournis pour les facteurs d'intensité de contrainte relatifs à une fissure alignée dans une direction quelconque par rapport à un point de la charge appliquée. On considère diverses approximations et on montre qu'une approximation d'une fissure courte simplifiée permet d'assurer une exactitude pour des valeurs de /a0,3 ( est la longueur de la fissure et a le rayon du disque). Pour certains angles, on trouve que la fermeture de la fissure peut se produire et en première étape, on prend en compte l'effet de la friction pour de tels cas en utilisant cette approximation de fissure courte. Enfin, des résultats expérimentaux sont présentés en vue de confirmer et élucider l'utilisation de cette géométrie d'éprouvette dans le problème d'un mode de rupture combinée.

     

Statistical analysis of the effects of prior load on fracture

Collated fracture data for three steels, A533B, A508 and BS1501, and an Aluminium alloy, 2650, were examined to assess the statistical significance of the effect of prior loading on subsequent fracture. Weibull statistical and probabilistic analysis was used throughout. Two prior loading conditions were examined; one associated with warm pre-stress, and a second using pre-compression. For the former, prior loading resulted in an increase in the mean toughness together with an increase in the shape parameter and a decrease in variability compared to the as-received material. In contrast, when prior loading involved out-of-plane compressive loading (or side-punching) statistical evidence revealed that there was a reduction in toughness together with a decrease in the shape parameter and an increase in variability. Two numerical models were applied to steel data and were able to predict the overall trends obtained from the experiments, but could not reproduce accurately the experimental statistical distributions.