Scattering of antiplane shear wave by a kinked crack

In this paper the scattering of antiplane shear waves by a kinked crack for a linearly elastic medium is considered. In order to solve the proposed problem, at first the broken crack problem is reduced to two coupled single cracks. Fourier integral transform method is employed to calculate the scattered field of a single crack. In order to derive the Cauchy type integral equations of a broken crack and analyze the singular stresses at the breakpoint, the scattered field of a single crack is separated into a singular part and a bounded part. The single crack solution is applied to derive the generalized Cauchy type integral equations of a broken crack. The singular stress and singular stress order are analyzed in the paper and the dynamic stress intensity factor (DSIF) at breakpoint is defined. Numerical solution of the obtained Cauchy type integral equations gives the DSIF at the crack tips and at the breakpoint. Comparison of the present results in some special cases with the known results confirms the proposed method. Some typical numerical results and corresponding analysis are presented at the end of the paper.     

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.

---

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.

---

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.

---

---

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

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.     

Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate

The problem of an anti-plane interface crack in a layered piezoelectric plate composed of two bonded dissimilar piezoelectric ceramic layers subjected to applied voltage is considered. It is assumed that the crack is either impermeable or permeable. An integral transform technique is employed to reduce the problem considered to dual integral equations, then to a Fredholm integral equation by introducing an auxiliary function. Field intensity factors and energy release rate are obtained in explicit form in terms of the auxiliary function. In particular, by solving analytically a resulting singular integral equation, they are determined explicitly in terms of given electromechanical loadings for the case of two bonded layers of equal thickness. Some numerical results are presented graphically to show the influence of the geometric parameters on the field intensity factors and the energy release rate.     

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.

---

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.

     

An interface crack in a functionally graded piezoelectric bi-layer under anti-plane shear impact

The transient response of an interface crack between two dissimilar functionally graded piezoelectric material (FGPM) layers under anti-plane shear impact loading is analyzed using the integral transform method. The properties of the FGPM layers vary continuously along the thickness, and the two layers are connected weak-discontinuously. Laplace transform and Fourier transform are used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate are presented for the FGPM to show the effects on the electric loading, variation and gradient of material properties, and thickness of layers. Following things are helpful to increase the resistance of transient fracture of interface crack in FGPMs: (a) increase of the material properties from the interface to the upper or lower free surface; (b) decrease of weak discontinuity at the interface; (c) increase of the gradient of material properties; (d) certain direction and magnitude of the electric loading; and (e) increase of the thickness of the FGPM layer.     

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.     

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.

---

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.

---

---

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

Dynamic linear elastic crack propagation in anti-plane shear by finite differences

The dynamic propagation of a crack in an anti-plane shear deformation field is analyzed by second-order-accurate finite differences. The finite difference equations are obtained by integrating the dynamic linear elastic equations of motion along the bicharacteristic strips in four perpendicular directions and the time axis to 0(t ³). The singularity in stresses around the crack is calculated by performing a global energy balance on small region containing the crack tip and approximating the stresses and velocity in this region by a one term asymptotic expansion about the crack tip. Results for stresses and stress intensity factor are presented for a semi-infinite crack propagating steadily in an infinite strip, from which errors in the numerical calculations are identified. Four cases of typical non-steady crack propagation in an infinite strip following steady propagation are also considered.

---

Résumé La propagation dynamique d'une fissure dans un champ de déformation de cisaillement anti-planaire est analysée en utilisant une méthode de différence finie exacte au second ordre. Les équations de différence finie sont obtenues en intégrant les équations de mouvement dynamique linéaire et élastique le long de 4 bandes caractéristiques distribuées dans 4 directions perpendiculaires et en fonction d'un axe de temps. La singularité des contraintes autour de la fissure est calculée en procédant à un équilibre global d'énergie dans les petites régions qui contiennent l'extrémité de la fissure, et en évaluant les contraintes et la vitesse dans cette région à l'aide d'une expansion asymptotique à un terme dans la zône de l'extrémité de la fissure. Les résultats obtenus pour les contraintes et le facteur d'intensité de contrainte sont présentés dans le cas d'une fissure semi-infinie qui se propage de manière stable dans une bande infinie, un cas pour lequel les erreurs des calculs numériques sont identifiées. Quatre cas de propagation typique de fissure non stable dans une bande infinie suivie d'une fissure stable sont également pris en considération.

     

The solution of a rectangular orthotropic sheet with a central crack under anti-plane shear

Making use of the basic theorem of the Fourier transform and series, the solution of the stress intensity factor of a rectangular orthotropic plate containing a central crack under anti-plane shear, is obtained in this study. The result to the mixed boundary value problem is expressed in terms of a Fredholm integral equation of the second kind. It is easily proved that the problem of a strip with a central crack of mode III, are the special cases of the solution in this article.     

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.

---

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.

---

---

With 3 Figures     

Transmission of anti-plane shear waves past an interface crack in dissimilar media

The scattered field generated by horizontally polarized incident shear waves in a system of layers having a flaw at the interface is studied. The physical model is idealized to the case of a crack sandwiched between two dissimilar media of infinite height. The local intensification of the dynamic stresses due to the crack is analyzed for incident waves directed at an arbitrary angle. Results of numerical computations are obtained by solving pairs of coupled integral equations and reveal the variations of the stress and displacement fields with ratios of densities and shear moduli of the two adjoining materials.     

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.     

Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials

In the present paper, the dynamic behavior of a Griffth crack in the functionally graded piezoelectric material (FGPM) is investigated. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPM vary continuously as an exponential function, and that FGPM is under the anti-plane mechanical loading and in-plane electrical loading. By using the Fourier transform and defining the jumps of displacement and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties on the stress and the electric displacement intensity factors.     

The kinked interface crack

The integral equation of the kinked interface crack is solved numerically. The values of K _I, K _II and G for an interface crack with an infinitesimal kink are used to predict the kinking angle for two different material combinations under uniaxial tension.     

Modeling fracture in the context of a strain-limiting theory of elasticity: a single anti-plane shear crack

This paper is the first part of an extended program to develop a theory of fracture in the context of strain-limiting theories of elasticity. This program exploits a novel approach to modeling the mechanical response of elastic, that is non-dissipative, materials through implicit constitutive relations. The particular class of models studied here can also be viewed as arising from an explicit theory in which the displacement gradient is specified to be a nonlinear function of stress. This modeling construct generalizes the classical Cauchy and Green theories of elasticity which are included as special cases. It was conjectured that special forms of these implicit theories that limit strains to physically realistic maximum levels even for arbitrarily large stresses would be ideal for modeling fracture by offering a modeling paradigm that avoids the crack-tip strain singularities characteristic of classical fracture theories. The simplest fracture setting in which to explore this conjecture is anti-plane shear. It is demonstrated herein that for a specific choice of strain-limiting elasticity theory, crack-tip strains do indeed remain bounded. Moreover, the theory predicts a bounded stress field in the neighborhood of a crack-tip and a cusp-shaped opening displacement. The results confirm the conjecture that use of a strain limiting explicit theory in which the displacement gradient is given as a function of stress for modeling the bulk constitutive behavior obviates the necessity of introducing ad hoc modeling constructs such as crack-tip cohesive or process zones in order to correct the unphysical stress and strain singularities predicted by classical linear elastic fracture mechanics.