Dynamic penny-shaped crack in a transversely isotropic material

The scattering of a harmonic longitudinal wave by a penny-shaped crack in a transversely isotropic material is investigated using the techniques of Hankel transform. The wave impinges normally on the crack surfaces. A complete contour integration is employed to simplify the expressions of the results. An exact expression of the dynamic stress-intensity factor is obtained as a function of the frequency factor and the anisotropic material constants. The normalized dynamic stress-intensity factor is shown to have different maximum values at different wave frequencies for the sample composite and metallic materials. The distortion of the dynamic crack shape and the displacement at the crack center are also shown to be dependent of the wave frequency and the anisotropy of the material.     

A transversely isotropic composite with a penny-shaped crack

Summary We consider the problem of determining the stress intensity factor and the crack energy in a transversely isotropic composite medium, containing a penny-shaped crack. We assume that the crack surface is perpendicular to the bond face and the crack is opened by constant internal pressure. By use of integral transform, we reduce the problem to solving a Fredholm integral equation of the second kind. Numerical results are given for the combination of some practical materials such as magnesium and cadmium. The effect of transverse isotropy upon the stress intensity factor, the crack energy and the deformation on the crack surface is discussed.

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Ein transversal, isotropes, komposites Medium mit einem münzenförmigen Riß

Zusammenfassung Das Problem der Bestimmung des Spannungsintensitätsfaktors und der Rißenergie, in einem transversalen, isotropen, kompositen Medium mit einem münzenförmigen Riß, wird betrachtet. Es wird vorausgesetzt, daß die Rißoberfläche normal zur Verbundfläche liegt, und der Riß sich durch konstanten inneren Druck öffnet. Durch Anwendung einer Integraltransformation, wird das Problem auf die Lösung einer Fredholmschen Integralgleichung zweiter Art reduziert. Numerische Ergebnisse werden für die Kombination einiger Materialien, wie Magnesium und Cadmium angegeben. Der Einfluß der transversalen Isotropie auf den Spannungskonzentrationsfaktor, die Rißenergie und die Deformation an der Rißoberfläche werden diskutiert.

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With 3 Figures

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This work is supported by the Board of Scientific and Industrial Research, Orissa (India).     

Plastic deformation in a thick transversely isotropic layer containing a penny-shaped crack

Summary The width of a thin plastic annular zone formed during the deformation of a pennyshaped crack in a transversely isotropic layer of an ideal elasto-plastic material is determined. Considered are the cases where the penny-shaped crack is extended by normal stresses and by torsional stresses. The faces of the layer are shear free and deformation of the plastic zone around the penny-shaped crack occurs according to the Dugdale hypothesis. For each case, the solution of the problem is reduced to a Fredholm integral equation of the second kind. Iterative solutions are obtained for small values of the parameters and numerical results for the width of the plastic zone are determined. Graphical results showing the effect of transverse isotropy upon the width of the plastic zone are also presented.With 6 Figures     

Point temperature solution for a penny-shaped crack in an infinite transversely isotropic thermo-piezo-elastic medium

The solution of an impermeable penny-shaped crack subjected to a concentrated thermal load (prescribed point temperature) applied arbitrarily at the crack surfaces is derived using the generalized potential theory method. The integral equation governing the temperature field is found to have the same structure as that for the elastic punch problem and the integro-differential equations related to the electroelastic field are similar to that reported for the elastic crack problem. Significant solutions to these integro-differential equations are obtained by generalizing the previous results available in literature. Exact three-dimensional expressions for the full-space thermo-electro-elastic field are finally obtained by simple differentiation, all in terms of elementary functions. The exact analysis for a permeable crack is also presented and discussed. The obtained point temperature solutions play an important role in the related BEM analysis.     

Singular stresses in a transversely isotropic circular cylinder with circumferential edge crack

This paper concerns the analysis of the singular stresses arising in a transversely isotropic infinite cylinder having a circumferential edge crack. The problem is reduced to that of solving a singular integral equation of the first kind which is solved numerically by the use of the way proposed by Erdogan, Gupta and Cook[1]. The singular stresses are expressed in closed form and the influence of transverse isotropy upon the stresses is clarified numerically for some practical materials.     

Dynamic stress field for torsional impact of a penny-shaped crack in a transversely isotropic functional graded strip

The torsional impact response of a penny-shaped crack in a transversely isotropic strip is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Investigated are the effects of material nonhomogeneity and orthotropy and strip’s highness on the dynamic stress intensity factor. The peak of the dynamic stress intensity factor can be suppressed by increasing the shear moduli’s gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface. The dynamic behavior varies little with the increasing of the strip’s highness.     

Penny-shaped crack in a transversely isotropic solid

An analytical solution is given for the displacement and stress distribution produced in the interior of a transversely isotropie solid containing a penny-shaped crack situated in an elastic symmetry plane and axially-loaded. Curves of numerical results are presented for the stress intensity factor and the normal displacement. They show the influence of this type of anisotropy.     

Dugdale model solutions for a penny-shaped crack in three-dimensional transversely isotropic piezoelectric media by boundary-integral equation method

Making use of the Displacement Discontinuity Boundary Integral Equation Method (DDBIEM), the dimension of the plastic zone at the tip of a penny-shaped crack in a three-dimensional elastic medium is determined by the application of the Dugdale model; Furthermore, the solutions for a penny-shaped crack in three-dimensional piezoelectric media are obtained by the use of the Dugdale-like model proposed by Gao et al.[Gao H, Zhang T, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 1997;45:491–510], in which the electrical polarization is assumed to reach a saturation limit in a thin annular region in front of a crack while the mechanical stresses have the ordinary singularity.     

A note on penny-shaped cracks in transversely isotropic materials

This note contains some further discussion of the problem of a penny-shaped crack in a transversely isotropic solid. For uniform applied stress at infinity, the problem is solved using Eshelby's method. Particular attention is given to the interaction energy and the crack opening displacements. The results are given in a form which is convenient in the study of cracked solids.     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium

This paper considers the non-axisymmetric three-dimensional problem of a penny-shaped crack with permeable electric conditions imposed on the crack surfaces, subjected to a pair of point normal forces applied symmetrically with respect to the crack plane. The crack is embedded in an infinite transversely isotropic piezoelectric body with the crack face perpendicular to the axis of material symmetry. Applying the symmetry of the problem under consideration then leads to a mixed–mixed boundary value problem of a half-space, for which potential theory method is employed for the purpose of analysis. The cases of equal eigenvalues are also discussed. Although the treatment differs from that for an impermeable crack reported in literature, the resulting governing equation still has a familiar structure. For the case of a point force, exact expressions for the full-space electro-elastic field are derived in terms of elementary functions with explicit stress and electric displacement intensity factors presented. The exact solution for a uniform loading is also given.     

Some crack problems in transversely isotropic solids

Summary. Crack problems in transversely isotropic solids are reexamined from a new point of view. It is shown that, when the crack is on the isotropic plane, the asymptotic forms of the elastic crack-tip fields are identical with those in orthotropic media. The equivalent inclusion method in conjunction with Eshelbys S tensor of a strongly oblate spheroid in transversely isotropic materials is used to solve penny-shaped crack problems. The stress intensity factors corresponding to uniform tension and shear are determined, respectively. Griffiths energy criterion for brittle cracking and Irwins energy release rate are discussed in the present context. Finally, the weight function for an axisymmetrically loaded penny-shaped crack is derived. It is found that the axisymmetric weight function is independent of the material constants and is identical with the isotropic case.AcknowledgementThis work was supported in part by the National Science Council of Taiwan.     

Dynamic steady-state crack propagation in a transversely isotropic viscoelastic body

The problem considered herein is the dynamic, subsonic, steady-state propagation of a semi-infinite, generalized plane strain crack in an infinite, transversely isotropic, linear viscoelastic body. The corresponding boundary value problem is considered initially for a general anisotropic, linear viscoelastic body and reduced via transform methods to a matrix Riemann–Hilbert problem. The general problem does not readily yield explicit closed form solutions, so attention is addressed to the special case of a transversely isotropic viscoelastic body whose principal axis of material symmetry is parallel to the crack edge. For this special case, the out-of-plane shear (Mode III), in-plane shear (Mode II) and in-plane opening (Mode I) modes uncouple. Explicit expressions are then constructed for all three Stress Intensity Factors (SIF). The analysis is valid for quite general forms for the relevant viscoelastic relaxation functions subject only to the thermodynamic restriction that work done in closed cycles be non-negative. As a special case, an analytical solution of the Mode I problem for a general isotropic linear viscoelastic material is obtained without the usual assumption of a constant Poissons ratio or exponential decay of the bulk and shear relaxation functions. The Mode I SIF is then calculated for a generalized standard linear solid with unequal mean relaxation times in bulk and shear leading to a non-constant Poissons ratio. Numerical simulations are performed for both point loading on the crack faces and for a uniform traction applied to a compact portion of the crack faces. In both cases, it is observed that the SIF can vanish for crack speeds well below the glassy Rayleigh wave speed. This phenomenon is not seen for Mode I cracks in elastic material or for Mode III cracks in viscoelastic material.     

Penny-shaped crack in a transversely isotropic plate of finite thickness

The effects of the material anisotropy on the stress intensity factor and on the crack shape are investigated for a penny-shaped crack in a transversely isotropic plate of finite thickness. The surfaces of the crack are subjected to uniform pressures. The plate surfaces are free from stresses for case I while smooth-clamp conditions are prescribed on the plate surfaces for case II. The techniques of Hankel transforms are used to obtain solutions for both cases. The solutions are largely written in terms of the sum and difference of the characteristic roots so that the results can easily be seen as real-value functions for both real and complex roots.Exact expressions for the stress intensity factor and the crack-shape function are obtained as products of dimensional quantities and nondimensional functions which are the stress intensity correction factor and the normalized crack shape function. The nondimensional functions were calculated numerically for three different typical materials which involved both real and complex characteristic roots. The numerical results clearly reveal the effects of the material anisotropy on the stress intensity factor and on the opening of the crack.

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Résumé On a étudié les effets de l'anisotropie du matériau sur le facteur d'intensité de contrainte et sur la forme d'une fissure dans le cas d'une fissuration en demi-lune située dans une plaque transversalement isotrope d'épaisseur finie. Les surfaces de la fissure ont été soumises à des pressions uniformes. Les surfaces de la plaque étaient libres de contrainte dans le cas I tandis que l'on prévoyait des conditions correspondant à un clamage léger sur les surfaces de la plaque dans un cas II. Les techniques de transformées de Hankel ont été utlisées pour obtenir les solutions dans les deux cas. Les solutions ont été exprimées en terme de somme et de différence de racines caractéristiques, de sorte que les résultats peuvent aisément être déduits comme des fonctions à valeur réelle de racine réelle et de racine complexe.Les expressions exactes pour le facteur d'intensité de contrainte et pour la fonction de forme de la fissure ont été obtenues comme les produits de fonctions à quantité dimensionnelle et non dimensionnelle qui sont le facteur de correction de l'intensité de contrainte et une fonction de forme de la fissure normalisée. Les fonctions sans dimension ont été calculées par voie numérique dans le cas de trois matériaux différents et typiques, mettant en oeuvre des racines caractéristiques réelles et des racines caractéristiques complexes. Les résultats numériques ont montré clairement les effets de l'anisotropie des matériaux sur le facteur d'intensité de contrainte et sur l'ouverture de la fissure.

     

Elastodynamic response of a penny-shaped crack in a cylinder of finite radius

The elastodynamic response of a penny-shaped crack in a cylinder of finite radius is investigated in this study. A step stress is applied to the crack surface resulting in transient behavior. The stress field near the crack front and the dynamic stress intensity factor are determined. Numerical resifits on the dynamic stress intensity factor are obtained to show the influence of inertia, geometry and their interactions on the load transfer to the crack.     

Punch and crack problems in transversely isotropic bodies

An exact solution is proposed for the mixed boundary-value problem in a transversely isotropic half-space. Here, certain arbitrary shear tractions are prescribed inside a circular region, outside of which certain arbitrary tangential displacements are given. The normal stresses are supposed to be known all over the boundary. A particular case is considered, in detail, where normal stresses vanish all over the boundary with the shear tractions vanishing inside the circular region. A closed form expression is obtained for the tangential displacements inside the circular region directly through the displacements outside. As an example, a penny-shaped crack in an infinite transversely isotropic body is considered with arbitrary shear tractions acting on both sides of the crack. The formulae for the tangential displacements inside the circle and the shear stresses outside are obtained. Special cases where uniform shear and a concentrated tangential force arise are also discussed.     

Dynamic crack problems in three-dimensional transversely isotropic solids

In this paper, a general boundary element approach for three-dimensional dynamic crack problems in transversely isotropic bodies is presented for the first time. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The procedure is based on the subdomain technique, the displacement integral representation for elastodynamic problems and the expressions of the time-harmonic point load fundamental solution for transversely isotropic media. Numerical results corresponding to cracks under the effects of impinging waves are presented. The accuracy of the present approach for the analysis of dynamic fracture mechanics problems in transversely isotropic solids is shown by comparison of the obtained results with existing solutions.