Stress distribution in the neighbourhood of Griffith and external cracks under general surface loadings

The problem of determining the stress distribution in an infinite elastic medium, containing Griffith cracks and the external cracks, is considered under the assumptions of plane strain. The crack surfaces are subjected to completely arbitrary surface tractions and the cracks are located on a straight line (say, x-axis).Detailed consideration is given to the cases when the elastic medium contains, (1) a Griffith crack ($\text{0 < ¦x¦ < a, y = 0}$); (2) a pair of symmetrically located Griffith cracks ($\text{0 < a < ¦x¦ < b, y = 0}$); and (3) a pair of external cracks ($\text{0 <a < ¦x¦, y = 0}$). The method of Fourier transform is used to analyse the problem and the results are compared, in special cases, with the known results in the literature.     

Penny shaped crack in a three-dimensional piezoelectric strip under in-plane normal loadings

Summary The singular mechanical and electric fields in a three-dimensional piezoelectric ceramic strip containing a penny shaped crack under in-plane normal mechanical and electrical loadings based on the continuous electric boundary conditions on the crack surface are considered here. The potential theory and Hankel transforms are used to obtain a system of dual integral equations, which is then expressed as a Fredholm integral equation. All sorts of field intensity factors of Mode I are given, and numerical values for PZT-6B piezoelectric ceramic are graphically shown.     

Eccentric crack in a rectangular piezoelectric medium under electromechanical loadings

Summary The solutions of an eccentric crack problem in a rectangular piezoelectric ceramic medium under combined anti-plane shear and in-plane electrical loadings are obtained by the continuous electric crack face condition. Fourier transforms and Fourier series are used to reduce the problem to two pairs of dual integral equations, which are then expressed by a Fredholm integral equation of the second kind. Numerical values of the stress intensity factor and the energy release rate are obtained to show the influence of the electric field.     

Stress intensity factors for penny and half-penny shaped cracks subjected to a stress gradient

Stress intensity factors at any point on the crack front of penny and half-penny shaped cracks subjected to stress gradients are presented. The SIF's which are exact for a penny shaped crack are based on the well known solution for a point load acting normally to such a crack. The line load solution which is derived from this is different in form to those given by previous workers and is more readily integrated to give SIF's for stress gradient loading. This is demonstrated by the derivation of a general equation for the SIF at any point on a penny-shaped crack due to polynomial stress gradients. These results are extended to produce a similarly general, albeit approximate, equation for the SIF at any point on the circumference of a half-penny crack due to polynomial loading. The usefulness of the approach developed here is further indicated by the derivation of an approximate SIF for exponential stress gradients over a half-penny crack.     

The penny crack beneath the surface of a half-space: with application to the blister test

The crack tip stress intensities are found for a penny shaped crack lying beneath the free surface of a half-space, and parallel with it. This is done by employing a novel distributed dislocation approach using axi-symmetric Somigliana dislocations as the kernel of an integral equation, and provides a precise solution at the expense of little computing cost. A comparison is made with the crack tip stress intensity factors for a simple plane crack, and predictions are made for the preferred crack extension direction.     

Transient response of a surface crack subjected to dynamic anti-plane concentrated loadings

In this study, the transient response of a surface crack in an elastic solid subjected to dynamic anti-plane concentrated loadings is investigated. The angles of the surface crack and the half-plane are 60° and 90°. In analyzing this problem, an infinite number of diffracted and reflected waves generated by the crack tip and edge boundaries must be taken into account and it will make the analysis extremely difficult. The solutions are determined by superposition of the proposed fundamental solution in the Laplace transform domain and by using the method of image. The fundamental solution to be used is the problem for applying exponentially distributed traction on the crack faces. The exact transient solutions of dynamic stress intensity factor are obtained and expressed in formulations of series form. The solutions are valid for an infinite length of time and have accounted for the contribution of an infinite number of diffracted waves. The explicit value of the dynamic overshot for the perpendicular surface crack is obtained from the analysis. Numerical results are evaluated which indicate that the dynamic stress intensity factors will oscillate near the correspondent static values after the first three or six waves have passed the crack tip.     

Elastodynamic stress intensity factors for a semi-infinite crack due to three-dimensional concentrated shear loadings on the crack faces

The dynamic stress intensity factors for a semi-infinite crack in an otherwise unbounded elastic body is investigated. The crack is subjected to a pair of suddenly-applied shear point loads on its faces at a distance l away from the crack tip. This problem is treated as the superposition of two problems. The first problem considers the disturbance by a concentrated shear force acting on the surface of an elastic half space, while the second problem discusses a half space with its surface subjected to the negative of the tangential surface displacements induced by the first problem in the front of the crack edge. A fundamental problem is proposed and solved by means of integral transforms together with the application of the Wiener–Hopf technique and Cagniard–de Hoop method. Exact expressions are then derived for the mode II and III dynamic stress intensity factors by taking integration over the fundamental solution. Some features of the solutions are discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

The stress intensity factor for a penny-shaped crack in an elastic body under the action of symmetric body forces

A formula is derived for the stress intensity factor at the rim of a penny-shaped crack in an infinite solid in which there is an axisymmetric distributing of body forces acting in a direction normal to the original crack surfaces. An expression for the surface displacement of the crack is also given. The use of these formulae is illustrated by a consideration of the special case in which the solid is deformed by the action of two point forces situated symmetrically with respect to the crack.

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Zusammenfassung Eine Formel für den Spannungsintensitätfaktor am Rande eines pfenninggeformten Risses in einem unendlichen Festkörper ist gewonnen. Ein achsensymmetrisches Verteilen der Körperkräite fand statt, welches in einer Richtung, normal zu der originalen Rissoberfläche wirkt. Es ist auch Ausdruck für den Oberflächenverschiebung des Risses gegeben. Die Benutzung dieser Gleichungen wird verdeutlicht durch die Betrachtung eines Spezialfalles bei dem der Festkörper durch die Wirkung zweier Punktkräfte deformiert wird, die symmetrisch zum Riss angebracht sind.

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Résumé On a établi une formule donnant le facteur de concentration de tension aux extrémités d'une fissure ferrnée disposée dans un solide infini au sein duquel une distribution de forces internes á symétrie axiale agit dans une direction normale par rapport aus surfaces de la fissure. On fournit également une expression du déplacement de ces surfaces. L'utilisation de ces formules est appliquée, à titre d' exemple, au cas spécial d'un solide soumis à l'action de deux forces concentrées symétriques par rapport à la fissure.

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This paper was prepared as a part of the work of the Applied Mathematics Research Group at North Carolina State University through the Grant AF-AFOSR-444-66 and is under the joint sponsorship of AFOSR, ARO, and ONR through the Joint Services Advisory Group.     

Fatigue crack growth simulation of aluminium alloy under spectrum loadings

Fatigue crack growth in structure components, which is subjected to variable amplitude loading, is a very complex subject. Studying of fatigue crack growth rate and fatigue life calculation under spectrum loading is vital in life prediction of engineering structures at higher reliability. The main aim of this paper is to address how to characterize the load sequence effects in fatigue crack propagation under variable amplitude loading. Thus, a fatigue life under various load spectra, which was predicted, based on the Austen, Forman and NASGRO models. The findings were then compared to the similar results using FASTRAN and AFGROW codes. These models are validated with the literature-based fatigue crack growth test data in 2024-T3 Aluminium alloys under various overload, underload, and spectrum loadings. With the consideration of the load cycle interactions, finally, the results show a good agreement in the behaviour with small differences in fatigue life compare to the test data.     

Dynamic stress distribution around the tip of a running crack

The stress distribution is obtained around the tip of a crack running in a brittle material. The stresses are written as the sum of the associated static solution and the wave-effect terms which depend upon the crack speed. The results obtained clearly reduce to the associated static solutions if the crack speed vanishes.Near the tip of the crack, the dynamic stress-intensity factor for the circumferential stress, $\text{σ}{}_{\mathrm{\theta \theta }}$, is written as the product of the associated static stress-intensity factor and the dynamic correction factor which is a nondimensional function of the crack speed, $\text{V}$, the angle from the crack plane, θ, and Poisson's ratio, ν. The value of the correction factor is computed for various values of $\text{V}$ and θ at $\text{ν = 0.25}$. It is shown that the maximum tensile value of $\text{σ}{}_{\mathrm{\theta \theta }}$, occurs on the crack plane for $\text{V}$ less than 0.7 time shear wave speed, $\text{c}{}_2$, and suddenly shifts to the plane of $\text{θ = 55°}$ for $\text{V}$ slightly larger than 0.7 $\text{c}{}_2$. For $\text{V > 0.7}$$\text{c}{}_2$, the angle θ for the maximum $\text{σ}{}_{\mathrm{\theta \theta }}$, θ being larger than 55°, varies continuously with the crack speed, $\text{V}$. The results obtained are used to discuss the growth of branching crack.     

Stress intensity factor at the surface and at the deepest point of a semi-elliptical surface crack in plates under stress gradients

The weight function method is used to calculate stress intensity factors for a semi-elliptical surface crack in a plate exposed to stress gradients. Starting from a reference load and stress intensity factor an approximate reference displacement field is calculated analytically. The present method allows to calculate stress intensity factors with minimal numerical effort at the deepest point and at the surface. Comparisons with FEM-results from the literature are presented to show satisfying agreement.     

Partial closure of a Griffith crack under a general loading

The plane problem of a Griffith crack, partially closed under the effect of a general loading, is considered. The three cases of: a) closure at one end under a nonsymmetrical loading; b) closure at the two ends under a symmetrical loading; and c) closure at the middle under a symmetrical loading, are considered separately. Examples are given from partial closures by concentrated forces of cracks opened by a uniform tension at infinity or a parabolic pressure on the crack surface. The effect of the concentrated forces in decreasing the stress intensity factors so as to prevent crack propagation is examined.     

Bridge-toughening analysis of the penny crack in a fiber reinforced composite with interfacial effect

A fracture mechanics analysis of bridge effect on a fiber reinforced composite containing a penny crack is presented. The integral equation governing bridge-toughening as well as crack opening displacement (COD) for the composite with interfacial layer is derived from the Castingliano's theorem and interface shear-lag model. A numerical result of the COD equation is obtained using iteration solution of the Fredholm integral equation of the second kind. In order to investigate the effect of various parameters on the toughening, an approximate analytical solution of the equation is presented and its error analysis is performed, which demonstrated the approximation solution to be appropriate. A parametric study of the influence of the length, interfacial shear modulus, thickness of the interphase, fiber radius, fiber volume fraction and properties of materials on composite toughening is therefore carried out.     

Calculating stress intensity factors for a semi-elliptical surface crack in a heterogeneous stress field

The author proposes an equation for calculating the stress intensity factor (SIF) for a semi-elliptical surface crack for uniform, linear, and quadratic laws of variation of the load applied to its edges. The derivation of the equation is based on the well-known Newman—Raju solution for a bent plate. The distribution of the values of SIF along the crack front, obtained using the empirical equations, coincides with the results of calculations carried out using the finite element method (FEM).Translated from Problemy Prochnosti, No. 7, pp. 38–41, July, 1990.     

Note on the growth of a penny-shaped crack in a general uniform applied stress field

Energetic arguments are used to discuss the growth of a penny-shaped crack situated within an infinite solid which is subject to tensile and shear stresses that are respectively normal and parallel to the crack plane. The most favourable growth mode is that for which the circular periphery becomes an ellipse, such that there is no growth perpendicular to the direction of application of the. shear stress; the appropriate growth condition is derived and compared with that obtained by assuming the circular crack to expand uniformly.