An observation of crack propagation in anti-plane shear

It is shown experimentally that in the absence of plasticity crack propagation in anti-plane shear occurs by the opening of semi-penny-shaped cracks which straddle the crack front at an angle of 45 degrees. It follows that if crack instability calculations are based on the assumption that a planar crack propagates in its own plane they are valid only if all shear stresses vanish along the crack periphery, i.e., if K _III is zero along that boundary.

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Zusammenfassung Die Ausbreitung eines RiBes im drei-dimensionalen Körper unter Einwirkung von Scheerspannungen die der Rißperipherie parallel sind wird experimentell untersucht. Es wird gezeigt daß für nichtplastische Materialien die Ausbreitung durch halbkreisförmige, flache Riße geschieht, die einen Winkel von 45° mit der Rißperipherie bilden.Hieraus ergibt sich die Folgerung daß Rißunstabilitätsberechnungen, mit der Annahme daß der RiB sich in seiner ursprünglicher Ebene ausbreitet, nur dann berechtigt sind, wenn alle Scheerspannungen entlang der Rißperipherie verschwinden, d.h., wenn K_III da gleich null ist.

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Résumé On montre par voie expérimentale qu'en l'absence de plasticité, la propagation d'une fissure sous une sollicitation de cisaillement antiplanaire se produit par l'ouverture de fissures en demi-lune, qui tend a écarter le front de la fissure selon un angle de 45 degrés. 11 en résulte que, si les calculs d'instabilité de fissure sont basés sur l'hypothése qu'une fissure plane se propage selon son propre plan, ces calculs n'ont de validité que pour autant que l'état de contrainte s'annule a la périphérie de la fissure, c'est a dire que K_III est égal a zéro le long de celle-ci.

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This work was supported by the National Aeronautics and Space Administration Research Grant No. NGL-05002-005 GALCIT 120.     

Dynamic propagation of a finite crack in a micropolar elastic solid

Summary The dynamic propagation of a finite crack under mode-I loading in a micropolar elastic solid is investigated. By using an integral transform method, a pair of two-dimensional singular integral equations governing stress and couple stress is formulated in terms of displacement transverse to the crack, macro and micro rotations, and microinertia. These equations are solved numerically, and solutions for dynamic stress intensity and couple stress intensity factors are obtained by utilizing the values of the strengths of the square root singularities in macrorotation and the gradient of microrotation at the crack tips.     

A finite element investigation of stable crack growth in anti-plane shear

Although the anti-plane strain case is of minor practical value in engineering applications, such an idealization facilitates mathematical investigations of strain and displacement fields accompanying extending cracks. This paper presents finite element solutions to anti-plane strain crack propagation problems and contrasts the numerical results with available analytic solutions in an effort to assess the accuracy of the numerical procedures. The nature of dominant strain singularities for stationary and moving cracks, the question of stableversus unstable or catastrophic crack growth and the implications of various proposed fracture criteria are discussed.

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Résumé Bien que le case des dilatations antiplanaires soit d'une importance pratique mineure dans les applications de la construction, une telle idéalisation facilite les investigations mathématiques sur les champs de dilatation et de déplacement qui accompagnent des fissures en cours d'extension. Le mémoire présente des solutions par élément fini aux problèmes de la propagation de fissure sous des dilatations antiplanaires et fait apparaître le contraste entre les résultats numériques et les solutions analytiques disponibles, dans une tentative de faire valoir ou de constater la précision des procédures numériques. La nature des singularités déterminantes de dilatation dans le cas de fissures stationnaires et de fissures en mouvement, la question de la croissance catastrophique instable ou stable d'une fissure et les implications que les divers critères proposés de rupture contiennent sont discutées.

     

Impact response of a finite crack in a finite strip under anti-plane shear

The response of a through-the thickness crack with finite dimensions to impact in a finite elastic strip is investigated in this study. The elastic strip is assumed to be subjected to anti-plane shear deformation. Laplace and Fourier transform were used to formulate the mixed boundary value problem. The dynamic stress intensity factor and crack opening displacement are obtained as a function of time and the strip width to crack length ratio, h/a. The results indicate that the intensity of the crack-tip stress field reaches a peak very quickly and then decreases in magnitude oscillating about the static value. In general, the dynamic stress intensity factor is higher for small $\text{h/a}$. Similar behavior has also been found for the crack surface displacement.     

The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids

The present paper is concerned with an infinite slab containing a crack and deformed at infinity to a state of finite simple shear. The material of the slab is taken to be homogeneous, isotropic, elastic, and incompressible, and is further assumed to belong to a class of materials which admit nontrivial states of anti-plane shear. The analysis is carried out for the fully nonlinear equilibrium theory of finite elasticity. The stress field near the crack-tips is studied in detail; one of the special materials considered is such that the shear stresses near a crack tip remain bounded, despite the presence of unbounded displacement gradients. An analogy between the crack problem in finite anti-plane shear and the problem of transonic flow of a gas past a flat plate is pointed out and discussed.

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Résumé Le présent mémoire est relatif à une plaque infinie comportant une fissure et déformée à l'infini dans un état de cisaillement fini simple. Le matériau de la plaque est considéré comme homogène, isotrope, élastique et incompressible, et il est en outre supposé appartenir à une classe de matériau qui admet des états non triviaux de cisaillement anti-planaire. L'analyse est effectuée suivant la théorie d'équilibre complðement non linéaire de l'élasticité finie. Le champ de contrainte au voisinage des extrémités de fissure est étudié dans le détail; un des matériaux spéciaux considérés est tel que les forces de cisaillement au voisinage de l'extrémité d'une fissure demeurent liées en dépit de la présence de gradiants de déplacement non liés. Une analogie entre le problème de fissuration dans une situation de cisaillement anti-planaire et le problème de l'écoulement transonique d'un gaz au-delà d'une tôle plane est mise en avant et discutée.

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The results communicated in this paper were obtained in the course of an investigation supported under Contract N00014-75-C-0196 between the California Institute of Technology and the Office of Naval Research.     

Non-symmetric extension of a crack in an infinite elastic medium under anti-plane shear stress

The dynamic problem of non-symmetric extension of a crack in an infinite elastic medium, which is initially in a state of uniform anti-plane shear, has been considered. The problem of non-symmetric extension of a crack due to cohesive traction has also been treated. The method of analysis is based on the observation that certain field quantities show dynamic similarity. The results include expressions for the stress intensity factors at the crack tips and the rate of energy flux into the crack edges for problem I. Numerical calculations are carried out to obtain stress intensity factors and the rate of energy flux into the crack tips for problem I.     

Closed-form solutions for finite length crack moving in a strip under anti-plane shear stress

Summary In this paper exact expressions for the anti-plane dynamic stress distributions around finite length cracks propagating with constant velocity in infinitely long finite width strips are determined. Two cases of practical importance are investigated. Firstly, the lateral boundaries of the strip are clamped and displaced in equal and opposite directions, to produce anti-plane shear resulting in a tearing motion along the leading edge of the crack and, secondly, the lateral boundaries of the strip are subjected to shearing stresses. Employing Fourier transforms the solution of each problem is reduced to solving a pair of dual integral equations. Closed-form solutions of these integral equations are obtained leading to exact expressions for the stress intensity factors. Numerical results are presented in graphical form.

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Geschlossene Lösungen für einen Riß endlicher Länge, der sich in einem unter antiplaner Schubspannung stehenden Streifen bewegt

Zusammenfassung In dieser Arbeit werden exakte Ausdrücke für die antiplanen, dynamischen Spannungsverteilungen um Risse endlicher Länge, die sich mit konstanter Geschwindigkeit in einem unendlich langen Streifen begrenzter Breite ausbreiten, bestimmt. Zwei Fälle von praktischer Bedeutung werden untersucht. Erstens werden die Seitenränder des Streifens eingespannt und sowohl in dieser als auch in entgegengesetzter Richtung versetzt, um einen antiplanen Schub zu erzeugen, der eine Aufreißbewegung längs der Vorderkante des Risses bewirkt und zweitens werden die Seitenränder des Streifens einer Schubspannung unterworfen. Die Lösung jedes Problems wird durch die Verwendung der Fouriertransformationen auf die Lösung zweier dualer Integralgleichungen reduziert. Es werden Lösungen dieser Integralgleichungen in geschlossener Form erhalten, die auf exakte Ausdrücke für den Spannungsintensitätsfaktor führen. Numerische Ergebnisse werden in graphischer Form gezeigt.

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With 3 Figures     

On the dynamic propagation of an anti-plane shear crack in a functionally graded piezoelectric strip

Summary. In this paper, a finite crack propagating at constant speed in a functionally graded piezoelectric material (FGPM) is studied. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential gradients along the thickness of the strip, and that the strip is under anti-plane shear mechanical and in-plane electrical loads. The analysis is conducted on the electrically unified (natural) crack boundary condition, which is related to the ellipsoidal crack parameters. By using the Fourier transform, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, crack propagation speed, electric field, FGPM gradation, crack length, and electromechanical coupling coefficient. It reveals that there are considerable differences between traditional electric crack models and the present unified crack model.     

Analysis of a kinked crack in anti-plane shear

A semi-infinite kinked crack in anti-plane shear is analyzed. The problem is formulated using the Mellin transform, and solved by the Wiener-Hopf technique. A closed form solution for displacement is obtained, from which the stress intensity factor is calculated. Particular emphasis is put on the stress intensity factor as the kinked length approaches zero, where two limit processes (both the distance from the crack tip and the kinked length approaching zero) are involved. It is found that the stress intensity factor depends on the order of performing the two limit processes. The results are compared with those by previous researchers. Also the energy release rate for this problem is computed.     

Thermoelastic wave propagation in an elastic solid containing a finite crack

Investigated in this paper is the scattering of plane harmonic thermoelastic waves around the tip of a finite crack. Integral transform techniques are used to formulate the problem and reduce it to Fredholm integral equations of the second kind. The equations are solved numerically and the singular stress field near the crack tip is determined. In particular, the variation of the stress intensity factor with the frequency of the incoming wave is exhibited graphically. The peak in the magnitude of the stress intensity factor is of paramount interest in the application of fracture mechanics to thermal stress problems.     

Dynamics steady crack propagation of antiplane shear in an elastic perfectly plastic material

Abstract

A theoretical analysis of steady‐state crack growth in an elastic ideally‐plastic material under small‐scale yielding conditions has been carried out for anti‐plane shear. Asymptotic expansion method is used to construct the solutions for the region near the crack line. Exact solutions for the distribution of strain on the crack line within the primary active plastic zone is obtained. It is shown that the solution reduces to the correct asymptotic form as the crack speed approaches zero (quasi‐static) for any point on the crack line. The results are used to discuss the applicability of quasi‐static solutions to moving steady‐state situations. It is found that if the crack propagation speed is less than 0.1 of the shear wave speed, the quasi‐static solutions can be accurately approximated for the steady state solutions.     

Dynamic crack propagation analysis of orthotropic media by the extended finite element method

Dynamic crack propagation of composites is investigated in this paper based on the recent advances and development of orthotropic enrichment functions within the framework of partition of unity and the extended finite element method (XFEM). The method allows for analysis of the whole crack propagation pattern on an unaltered finite element mesh, defined independent of the existence of any predefined crack or its propagation path. A relatively simple, though efficient formulation is implemented, which consists of using a dynamic crack initiation toughness, a crack orientation along the maximum circumferential stress, and a simple equation to presume the crack speed. Dynamic stress intensity factors (DSIFs) are evaluated by means of the domain separation integral method. The governing elastodynamics equation is first transformed into a standard weak formulation and is then discretized into an XFEM system of time dependent equations, to be solved by the unconditionally stable Newmark time integration scheme. A number of benchmark and test problems are simulated and the results are compared with available reference results.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

Bonded wedges with an interface crack under anti-plane shear loading

The primary aim of this paper is to describe an analytical technique which may be used in connection with the general problem of bonded wedges containing radial cracks. The technique consists of the reduction of the related dual integral equations of the problem to a singular integral equation in a systematic manner, and is described by applying it to a relatively simple anti-plane shear problem. The paper also presents the results of various numerical examples and the closed form solution for the special case of two bonded wedges with equal angles and an interface crack.

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Résumé L'objet principal du mémoire est de décrire une technique d'analyse utilisable dans le cas du problème général de secteurs jointifs comportant des fissures radiales. La technique consiste à réduire de manière systématique à une équation intégrale singulière les équations intégrales décrivant le problème; un exemple d'application est donné dans le cas d'un problème relativement simple de cisaillement antiplanaire. L'article présente également les résultats de divers exemples numériques, et la solution de forme fermée que l'on trouve dans le cas particulier de deux secteurs collés ayant un angle au centre égal et une fissure dans leur surface de séparation.

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This work was supported by the National Science Foundation under the Grant GK-11977.     

Mixed finite element formulation for the general anti-plane shear problem,including mode III crack computations,in the framework of dipolar linear gradient elasticity

A mixed formulation is developed and numerically validated for the general 2D anti-plane shear problem in micro-structured solids governed by dipolar strain gradient elasticity. The current mixed formulation employs the form II statement of the gradient elasticity theory and uses the double stress components and the displacement field as main variables. High order, C ⁰-continuous, conforming basis functions are employed in the finite element approximations (p-version). The results for the mode III crack problem reveal that, with proper mesh refinement at the areas of high solution gradients, the current approximation method captures the exact solution behaviour at different length scales, which depend on the size of material micro-structure. The latter is of vital importance because, near the crack tip, the nature of the exact solution, changes radically as we proceed from the macro- to micro-scale.     

Dynamic crack propagation in composites

A study of the behaviour of cracked composite specimens under dynamic tensile load was undertaken. The crack propagation in the two-phase epoxy resin specimens was studied by the method of high speed photography along with the optical method of caustics.Our investigation was concentrated both on the dependence of the maximum crack propagation velocity and the stress intensity factor at the crack tip upon the different material combinations of the composite, as well as on the role of the interface again in regard to the crack propagation and the singular stress field concentrations at the crack tip.The results show that, under a given value of the applied dynamic load and given notch dimensions, the stress intensity factor at the crack tip and the crack propagation velocity in each phase of the composite is highly dependent on the material characteristics of each phase and on the existence of a stable interface between the two phases.More concretely, it was proved that the interface plays the role of a barrier to the crack propagation. Indeed, the crack propagates with a certain maximum velocity in the first phase of the composite and then stops momentarily when it reaches the interface, thus attaining later in the second phase a new maximum velocity.The maximum velocity and the stress intensity factor in the second phase of the composite specimens strongly depend on the material characteristics of the first (notched) phase and are also highly influenced by the crack arrest process itself.The crack propagation and the stress field concentrations at the crack tip in the first phase of the composite specimens is mainly independent from the material characteristics of the second phase of the composite specimens.

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Résumé On a entrepris une étude sur le comportement d'éprouvettes composites figurées sous contrainte de traction dynamique. La propagation de la fissure dans des éprouvettes de résine Epoxy à 2 phases a été étudiée par la méthode de photographie en grande vitesse ainsi qu'en utilisant la méthode optique des caustiques.L'étude a été concentrée à la fois sur la dépendance de la vitesse maximum de propagation d'une fissure ainsi que du facteur d'intensité d'entaille à l'extrémité d'une fissure sur les différentes combinaisons de matériaux pouvant constituer le composite et sur le rôle de l'interface existant entre la propagation de la fissure et les concentrations de champs de contrainte à l'extrémité de la fissure.Les résultats montrent que, sous des valeurs déterminées de la charge dynamique appliquée et pour des dimensions d'entaille donnée, le facteur d'intensité des contraintes à l'extrémité de la fissure et la vitesse de propagation de la fissure dans chacune des phases du composite dépend dans une large mesure des caractéristiques du matériau de chaque phase et de l'existence d'un interface stable entre les 2 phases.Concrètement parlant, on a pu établir que l'interface joue le rôle d'une barrière à la propagation de la fissure. En effet, la fissure se propage avec une certaine vitesse maximum dans la première phase du composite et ensuite s'arrête momentanément lorsqu'elle atteint l'interface jusqu'à reprendre sa course dans la deuxième phase du composite avec une nouvelle vitesse maximum.La vitesse maximum et le facteur d'intensité des contraintes dans la seconde phase de l'éprouvette composite dépend fortement des caractéristiques du matériau de la première phase (la phase entaillée) et sont également largement influencés par le processus d'arrêt des fissures lui-même.La propagation des fissures et les concentrations du champ de contrainte à l'extrémité de la fissure dans la première phase d'une éprouvette composite est généralement indépendant des caractéristiques du matériau de la deuxième phase de ce même échantillon.

     

Radial propagation of axial shear waves in an incompressible elastic material under finite deformation

Equations derived via Biot's mechanics of incremental deformations are employed as a basis for investigation of radial propagation of axial shear waves through an initially radially deformed elastic solid. In the first instance, reduction of the propagation equation is sought to one of the two canonical forms. One is soluble by finite Hankel transform techniques while the other is associated with the conventional one-dimensional wave equation. Two specific initial value problems involving a neo-Hookean solid are solved explicitly by means of these techniques. Finally, an alternative approach is presented which utilises formal asymptotic wavefront expansions.